I know I would not be able to do what you are doing. I would tell my H he either comes to counselling with me to deal with what I consider an unacceptable situation, or he moves out. But I'm not saying that's right, it's just what my knee-jerk reaction would be.
A friend of mine, her husband was seeing a younger woman from his work. She was a single parent and kind of a dependent thing. He was spending an inordinate amount of time with her, so my friend said listen up Bud, we're going for counselling or our marriage might be in the toilet. To his credit, he went to counselling and agreed to stop seeing the younger co-worker. My friend doesn't think they had a sexual relationship (who knows) but I am very impressed with how she dealt with things. The fact her husband responded to her, means he was able to carry out his commitment to her.
Your H wants to take responsibility for getting someone else pregnant by providing for the child, I think that's required by the law, but his first responsibility is to you and your boys. If he can't respond to your requests to go for counselling, how do you see this situation playing itself out? I think if you earn your own income, I would definitely have separate accounts because I wouldn't want any of my income going to her. On the other hand, I wouldn't feel comfortable with him having his own account to spend at will on her. Does he not have financial obligation to you being that he lives with you?
Aieeee! Hopenfaith, I would get him into counselling because any counsellor, christian or otherwise, will give him a reality check that this situation is entirely bizarre and unfair to you and your sons. He needs to make a choice.