I have known women whose marriages have come through affairs wonderfully-others not so much. Will he consider counseling? Mine said NO WAY for years. Are you receiving counseling? The counselor can help you work through your turmoil. Hoping the best for you and your children.

He has already left you psychologically with what he is doing. Do you have a minister or priest you can talk with & pray with about this?
It would bother me if husband was sleeping with another woman & wanted any intimacy with me, as it would not longer be safe (chance of diseases).
Not sure how you are holding up through all of this but you have every right to be upset.
I agree with Becky's advice, & I wouldn't do anything drastic but start talking & seeking supports & advice so your life has some order to it again. That's my suggestion!