What I often point out is that if a human being has a surfeit of needs being met, there is no hunger or motivation. However, people will often take extra if it is offered.
It's only when something becomes out of reach or unattainable that we want it more - there is a strong motivation to get that need met.
Getting your needs met this way is damaging to you, because competitive sex will mess with your head and when you look at it logically, you are two women competing for a very tarnished prize. Having sex when you need to detach is self-sabotage.
What you are also doing by meeting any affection or sexual needs in him is also sabotaging what you appear to want. He will only ever want you when he thinks he has lost you for good. You and the OW are propping eachother up in a way; you are both meeting his needs and have been for a long time.
Being objective (and I know you're not) the best goal in these situations is for the man to be left with no-one. But all the while you and the OW are competing with one another, he has a surfeit of needs and no motivation to change or let one relationship stand on its own two feet.
Because you still love him, it's obvious that you cannot be hard-hearted enough to stop doing things for him all the while he is biding his time and living under your roof. So he will be there and getting things done for him, all the while he is planning his escape.
This really will erode your mental health more than you can imagine right now. At the moment because you haven't looked into the finances, you've no idea whether you could actually be better off with him living elsewhere and contributing less because of his rent.
This stalemate is far too easy for him and will be hell for you. Can I suggest that you find out about the finances and any benefits and consider revoking your decision to let him stay? He really needs to be out in the cold, because he is planning a new life on your time and in your house.