Removal of the oaths would cause the entire world to focus their attention on the Aes Sedai, and they would then be forced to change their positions on a lot of things, and radically alter their organization from its current form into one more beneficial for the world, because otherwise the nations of the world would not accept unbound tyrants bossing them around. So no, they wouldn't be able to have secrecy anymore (to the extent they have now). Instead they'd become like a normal political institution, and have to actually have justifiable reasons for what they do, and not just their secret Aes Sedai reasons that are essentially them saying that only they can possibly understand what is good for the world. Over time if they proved themselves, people would stop fearing them so much, and then they could revert to their old ways without having to fear being slaughtered, because by then they'd be seen as actual human beings. But at the moment they've separated themselves too much, and humanity would not accept them unless they really went about showing how they've changed.

The Black Ajah would most certainly resurrect itself in time, probably a lot of time. But without the Oaths I think it would actually be harder for the Black Ajah to operate, because Aes Sedai would no longer accept as unassailable truth things that other sisters tell them. Instead they'll actually pay attention to what the others say and do in order to weed out the Black Ajah members, so the true Black Ajah would no longer have the protection that the assumption of their truthfulness has given them thus far.