I will deal with the two characters separately:

Egwene: I assume you mean her talks with the Sea Folk and the Wise Ones? I hope you're not saying that when she got control over the WT's dealings with Rand, she was doing so to get power. anyway, she knows as well as we do how useless that idea would be.

Now... what is wrong with Egwene wanting to tie the three channeling organizations together? All three of them have lost women to the Seanchan. All of them face an existential crisis if the Seanchan should succeed. They all, at one point, likely intend to strike at the Seanchan...

Further, for the Last Battle itself, it makes more sense that these women can trust each other as part of a larger sisterhood of channelers. If they each had a stake in the survival and prosperity of the others, then the natural suspicion with which they view each other is bound to diminish. That is definitely a great plus for the LB, and from Egwene's perspective, extremely critical to convince Rand that he shouldn't break the Seals.

I fail to see how this can be seen as a needless power trip in the face of the Last Battle.

Elayne: By taking Cairhein, in one stroke she handles several issues. First, Andor's feuds with that nation are necessarily at an end. Further, bickering nobles like Ellorien (and those Cairheinin), will have a tougher time opposing her since she is queen of both lands. Sure, she'll have to continue playing the game, but the benefits for the Last Battle are staggering. A united Two Rivers-Ghealdan-Andor-Cairhein can form a much stronger bulwark against the Shadow, and can also put up a much better fight against the Seanchan. The Seanchan are also the reason why Egwene will likely really want Elayne as Queen of Cairhein. Tar Valon's southern borders are now under a firm ally. The value of that against the Seanchan is incredible.

As for the Dragons... Elayne has every right to want them, and to want to keep them secret as long as possible. That gives her a much needed edge against the Seanchan. They cannot now simply roll over Andor, and add Aes Sedai to the mix and Andor will be a tough nut to crack indeed.

Essentially, both these women are getting ready for the Last Battle but also ensuring that after it is won, they don't end up collared slaves of the Seanchan empire.