Hi OP, I'm sorry you find yourself in this situation.
I really think you need to make this decision for him and call an end to this marriage.
I have been having an affair for just over 5 years, we were caught by his wife 8 times, she found out initially, then she found out we were still in contact, he woke her in the night once to say he didn't love her he loved me and was leaving but she talked him round, he did leave her briefly but went back and so on and so on. He done all the usual things, deleting my number .....but of course he knows it off by heart anyway so deleting it is pointless. I don't believe just because he's sworn in your child's life that he hasn't slept with her because I imagine he has. Sorry I'm being cynical but I lived it for so long I've heard all the excuses, seen all the tricks used. It sounds like he doesn't love you anymore but he doesn't want to lose his children so he stays out of fear. I wouldn't want anyone being with me if they were in love with someone else, it must be soul destroying.
My AP stays with his wife because of the history they have, because he doesn't want his relationship with his kids ruined, because he cares too much about what other people think of him, many many reasons but none are because he loves his wife. She has told him if he leaves her she will tell his kids various stuff so there's an element of fear there too. I feel sorry for her living like that, she checks his phone, tracks his movements, but ultimately she chose to stay with him knowing what had been going on so much as I feel for the way she lives, I think she's an absolute mug who is as weak as he is.
Your dh wants you to make the decision for him, and whilst he has feelings for this other woman he will never commit to your marriage because his head and heart are elsewhere. The more controlling you are the more he is going to resent you and even if him and this other lady are not suited, in his mind they will be like Romeo and Juliet, with you being the obstacle that stops him being with you. Not the recipe for a happy life that's for sure.
Sorry if this is negative, but I've lived it from the other side of the fence and obviously me and AP spoke loads about his relationship and life with his wife and I learned loads from that.
Wishing you good luck and strength.