First off, I would make sure he has stopped all conversations with her. Also, if you don't trust him anymore(which I don't blame you, I wouldnt either), then there isn't any hope for your marriage to survive, you'll never be happy, if you can't get past this.
About your kids. YES as a mom you DO have a right. I am strongly against people staying together to benefit the kids. That is wrong and they might resent you for that if they have a bad childhood with you staying with your husband(their father) because of them. Its better to move on, allow them to have a relationship with their father, and you find someone that can make YOU happy!
maybe this woman being here in your lives, made you see that you and your husband wasnt meant to be together.