It's really very simple. To be able to leave this letter to Mat, Verin had to convnce herself he wouldn't open it before her last hour or after she rid herself of the BA oaths. Almost certainly, she had doubts if she simply ordered him not to read the letter before x days, curiosity might get the better of him, and those doubts would have made it impossible for her to give him the letter. She managed to convince herself that Mat would never open the letter before time if she made him promise to obey orders he wouldn't know before opening the letter, so he would choose to open the letter only once he was no longer bound to obey anything, And that let her circumvent her oath. Verin had no doubt Mat was the right person and once he read the letter he would act - it was never about binding Mat to follow her information,it was always about finding a way around her oaths. She gave a letter to Galad, someone widely known to always keep his word. Is the letter really for him, or is he just a safekeeper and courrier, meant to pass information to others later? We'll see. Alanna wasn't a problem if Verin has ruled out she could be BA, as she could get a binding promise out of her - she wouldn't be able to open the letter too soon once she promised not to. Verin's choices of recipients was limited, it had to be people with whom she had ways to circumvent her oaths. Otherwise, she would have needed to place wards on the letters so they would be destroyed if someone opened them too soon, which meant risking her information would be lost. It's likely a device like this she would have needed to leave a letter to Elayne, for instance, or using someone she could trust to obey an order not to send the letters to their final recipients before her death. But doing that was high risk. If something happened to that person, a lot of importsnt letters would be lost at once, and that's not accounting for the risks to reach many persons would require birds, and birds can bet lost or intercepted.

Why not tell Egwene? She could not be sure she would reach her or that she'd succeed with her, the letters made sure she would not completely fail if she failed with Egwene. For the rest, she wasn't about to overload Egwene with urgent information she must act on when Egwene was the one she had chosen to bring down the BA - she had more than enough on her plate with just that, especially that Verin knew Egwene would have to work hard from the position she was in if she were to succeed. Verin was also not aware that Egwene could have information passed to Elayne easily. And she had no reason to believe Mat wouldn't open her letter as soon as the delay during which he would have to follow orders was over, so the Caemlyn situation was taken care of for Verin.