Why is this specific to this situation, from you? Where was your outrage at Rand flinging fire and destruction at the Shaido in tFoH, instead of just trapping them in Air? We know that at that point, he could trap multiple people in a single web of Air, and also make walls of Air. Yet there he was, burning off entire hilltops. The Shaido surely deserve as much consideration as Sul'dam, right?

In a combat situation, tying them up with Air, then having them transported somewhere safe would add to the danger to herself and her followers, since she needs everyone focussed on repelling the Seanchan. Unlike the Damane, the Sul'dam are NOT going to be passive and too scared to take initiative as a rule.

Depends on the Damane, and Egwene seems aware of the risk:

You see, she admits she took a greater risk for whatever reason in this case. If she can take the risk to spare the occasional damane, she can take the much lesser risk of not murdering unarmed non-channelers, especially when she is supposed to be setting an example to novices.

This is the really annoying thing about Sanderson. So much of his writing derails conversations about the series, because we can't be sure if something happened, but was not mentioned properly, or was his own initiative in mentioning it, but was out of character and a fan-fic moment he thought would be cool, or something else stupid.
He was not so positioned, and they were in position to be a threat to other people. There is no proof that he could do his Air trick at any great distance, nor were they under his direct control, being so far away. Note that when he IS up close and personal, and no follower's lives were at stake (except for Aviendha, but it was her fault they were in that situation in the first place) he DOES immobilize the Seanchan, and does not kill anyone he does not need to.
Rand NEVER murders anyone in close quarters like Egwene does. A pitched battlefield is completely different from a one-on-one encounter.
Name someone else who did the same thing. Seriously. They don't. This is why I "single out" Egwene, because she is repeatedly the only one who finds herself in such situations. Aside from maybe a background Asha'man, she is the only person to directly express a desire to use the Power as a weapon. Nynaeve has some nebulous plans to use the Power against Moiraine in the future to avenge her ruining the lives of the Two Rivers folk, but it is only Egwene who ever WANTS to lash out with the Power, it is only Egwene who seeks opportunities to use the Power even when she is insanely outmatched, who must always be forced to exercise discretion. And Egwene is the only one to ever, in a personal combat situation, murder helpless people like that. I didn't call her out for her bloodlust during Tarmon Gaidon, and I have defended her killing the to'raken in the same fight in Tar Valon, because the same circumstances apply to her as with Rand at Cairhien.
Yeah, no Aes Sedai would do a thing like that, only Egwene ever attacks the Children with the Power.

BTW, remember, the Seanchan were counter-attacking. They might have had the wrong idea about the White Tower's role or willingness to protect the portion of the world over which they claimed influence, but as we see in WoT, moving out of a mistaken understanding of a situation is simply human nature.

But being part of such an attack does not forever strip you of the right to basic human decency.
So far as they know. The Children have no way of verifying the truth of the Oaths, and we see frequently how much of a sham those are when an Aes Sedai wants to get around them.
I never said she had to do that for the sul'dam. There is a whole gulf of options between coddling and burning alive.
They commented on it in just about every book, that Egwene kept harping on the necessity for living according to the Oaths, even before that conversation. That was one of those off-screen conversations that somehow happen, but we never get to see, like Egwene's apparent conveyance of Elayne's feelings to Rand. We see that happen in one conversation, but Rand mentions a constant stream of well-wishings, and Egwene uses further updates as an incentive for him to want her back in T'A'R. That conversation where Egwene lays down the law for good starts off in a way that plainly indicates this is not their first go-round on the topic, since they are basically asking her if she has changed her mind yet. And as the specific and primary beneficiary of that political importance, it is even further incumbent upon her to stick to them. Siuan's point was that political unity imparted by the Oaths, which benefits Egwene more than anyone else, since unity means more followers under her rule.
True.

See, this is one place where I err on the side of giving Egwene the benefit of the doubt, to a certain degree. As far as I am concerned, the political motivation is so petty and contemptible and dishonest, that I find it less disgust-inducing to give her credit for the implied moral impulses behind which the Oaths disguise themselves. There are a few times when she seems to appreciate the First Oath out of respect for a version of honesty. I'd rather give her credit for some limited and flawed impulses toward honesty and non-violence, than accuse her of paying lip service to a morally bankrupt code of conduct solely to gain the dishonest advantages of the false reputation it imparts. In spite of all logic and rationality, much less knowing the outcome of her arc, I still kind of cling to the image of her as the Tower's best hope for reform and rehabilitation, which of course, just sets me up to be disappointed when she throws each opportunity away.
They are no such thing. That has massive loopholes for anyone with half a brain, and they can sic their Warders on people in any case. The point of the Oaths is to give the Aes Sedai an out and to prevent them from committing themselves to any cause, while gaining the benefits of alliances and dodging any reciprocal obligations they find inconvenient. That they are an effective tool in binding the Tower together says nothing good about the sisters, or their susceptibility to brainwashing.
The only major character to ever, at any point, advocate such a position, was Egwene's defense of Aram when Perrin pointed out the obvious flaws in the Way of the Leaf. I also never advocated non-violence, but again, there is a big gap between that and burning helpless people alive.

That she thinks she has the right to such expectations is, in the abstract, more horrifying than her frequent impulse to settle a situation in the bloodiest way possible, or desire to get involved in fights that don't concern her, or where any other character would be criticized for overreaching his capabilities.

It's not a plan, it's a bizarre and unworkable notion to paper over a serious objection to her intention to retain a political advantage at the expense of her own expressed desire to institute a necessary reform. Aside from the annoying lifespan issues, these women are still supposed to take Oaths to act according to a dishonest set of rules. An oath, and the responsibility to keep it, is not cancelled out by the opportunity to do otherwise. Just as appalling as Siuan getting away with her lie about the Reds and the False Dragons, is that she swore an oath not to lie and lied at the very first opportunity she had! It's like swearing a vow of chastity and then having sex with the very first person you encounter. And she told her first lie and most subsequent ones, with the full knowledge that the people to whom she lied believed she was under oath.

Anyway, Egwene is either going to be putting these future Kinswomen in an impossible position, as well as retaining that artificial distinction between Aes Sedai (and now all channelers) and normal people, or else she's going to be making a mockery of the whole point of the Oaths, by having an association of the Tower running around unbound.

Hence the problem with the Oaths. Permitted by the Oaths (or the Aes Sedai's mental shortcomings) does not make something right. Note that shortly before the fight there is a similar issue, where she wishes for the Oath Rod because of the necessity of explaining Verin's presence in her bed without letting news of her death get out. She tells an untruth, that is terminologically precise, is still intended to mislead, while wishing for the Oath Rod, since lying is too convenient and tempting in such times. Egwene plainly aspires to more than the politically convenient lip-service to an otherwise meaningless set of Oaths, while at the same time, missing the point of the moral rule, and substituting the standard of the Oaths. Just because the words she uses to Turese are technically true does not mean she is not deliberately using them with intent to deceive. That is a lie, just as surely using false words. That's why the oath of a witness forbids lies of omission ( "tell...the whole truth..." ) and mixing true statements with false to conceal the latter ( "...nothing but the truth" ). Lying is not always wrong, except when attempting to mislead others about the truth of another, but the Oaths change the moral axis to allow lies, so long as they can be told by use of facts. And Egwene indicates she is accepting the Oath's version of honesty in place of the morality she once tacitly accepted. The so-called lie for wish she abuses Nynaeve in T'A'R as punishment, would have been completely acceptable according to the First Oath. Nynaeve's words were more or less "imagine trying to give forkroot to someone who knows herbs like I do." She never denies drinking the tea or anything that happened, nor does she assert the occurrence of something that did not happen. Yet, when she confesses, Egwene still acts as if it is a lie and frames it as a personal betrayal. So something has plainly changed by the time she has come back to the Tower.

We see too many abuses of the Third Oath, covered over by the ignorance or timidity of the sisters in question to assert that burning the sul'dam alive would have been impossible had Egwene been sworn on the Oath Rod. That does not remotely make it right, any more than Coiren acknowledging that they owe Sevanna for services rendered, while deliberately evading a commitment to render restitution for that service, or that Kiruna was right in getting soldiers killed by deviating from the battle plan for the political convenience of the rebel embassy. Since her sisters and Alanna's warder were under attack by the enemy in question, she had all the clearance she could wish to blast the Shaido at will. No one doubts that political motivation, and she does not attempt to deny it, but that she can believe the excuse she offers for the 'necessity' of her actions, shows how the Oaths don't prevent sisters from being stupid or ignoring realities.

Just that at some point, it seemed like she aspired to the spirit of the moral underpinnings to which the Oaths pretended.
I would go more in the direction of "shows a propensity or inclination toward violence with the OP, particularly for self-gratifying reasons. Her impulsive attacks in Falme and outside Tar Valon, her repeated defiance on the topic of using the Power as a weapon during the journey to Tear, in defiance of the point Nynaeve made both verbally and by example, of non-lethal alternatives, combined with her emotional overreactions to the Seanchan, all point to an unnecessarily lethal option as in keeping with previously demonstrated character traits.

I am not even unsympathetic to the reasons for her animus toward the Seanchan. But that is not something she has the luxury of retaining if she aspires to leadership.

She did. And I see you've grasped the "whole thing" idiom.