You know your WAW best, but I would suspect she might want to see if another woman is sleeping there while the children are in the house, as a weapon against you legally? I know, she won't see the hypocrisy in that. I get the impression that your WAW goes over the top, to unbalance those she is attacking. Most people would be happy to have an amiable divorce with 50/50 child custody. But not her.
She doesn't seem to want to do things the easy way ... she wants you grovelling, without the children, for whatever reason. There's an itch there that she just wants scratched, and you're the post she's gonna do it on. (Think about her first husband and how she treated him during the D.)
Normally, I would say, yes, go for it. See what the problem is that caused her to suddenly try for full custody. Seek a friendly D with 50/50 custody. But, I think we all know why she's doing it .... she wants to take the children out of the country and she is resentful of you standing in the way, IMHO. She's throwing the dice, gambling with the possibility that she may win, and if she doesn't, nothing is lost (unless you sue for full custody, and she knows you are a reasonable person, who likely won't do that to her or the children.) I'm just not sure whether you will find out anything, and she won't capitulate unless she gets something out of it .... passports for the children and agreement to let them out of the country. I doubt if she will show her hand ... only what she wants you to see.
Remember that this is someone who feels no guilt over having an affair. You were not an evil H who beat her, isolated her, had multiple affairs. You tried to make her happy, and you love your children and you are a good dad. There is nothing she can say that justifies the A. And there is nothing she can say that justifies her having full custody of the children. If anything, you have more justification for that than she does, and she is lucky that you are a good man and don't want to deprive the kids of their mother. You just have to realize that she doesn't think the same way you do.
But, I end with what I began with ... you know her best. Only you can decide what the best course of action is. If you do approach her, I would make sure there is evidence, i.e. send an email and hope for a reply, or a letter (and make a copy), or have a witness present, so that you can show you tried to make it work outside the court system.
Take care.
Me:57 H:52 M:28 Got another lawyer last year and filed. D35,S/D twins28,D22 EA4/04 End? Who knows? "Life is like a mirror. Smile at it and it smiles back at you." — Peace Pilgrim