I would go country hardball on her legally. Use the leverage of being able to go after her for grounds of adultery, and put a sweetheart (for you and your boys) deal in front of her. Tell her it is not negotiable, and that if she wants to fight you on this, you will go after her with everything you have.
So she's wearing her wedding ring. While she cheats on you.
The current situation is not good for your kids. Time to go "Papa Bear," in my opinion.
