A friend of mine who is an accountant, advised me that it would be better to own the house outright now, and received less maintenance, that it is better in the long run, but of course you have to be able to survive financially in the meantime and I needed the maintenance, little though it was!. I was lucky, XH walked away from the equity as it was all mine before I met him, and he did the decent thing only decent thing he did do. and in my case, the solicitor was right, get him now, while he is saying these things and get it all tied up legally. When he changes his mind, he will fight you in the courts for every penny.
Your H can sign a deed that signs the ownership of the house over to you once his name is off the mortgage.
If he is paying more than CSA for now then let it go, but if he drops it below, then get straight on to the CSA.
As inlectorecumbit says do you have pensions? You could be entitled to a claim on his and vice versa. It is far better to get the financial side of things tied up as soon as possible, but if you feel that you cannot face divorce then that is your decision. I had to for financial reasons and XH's history of debt problems, but it was awful divorcing a man that I still loved, having thought we were happy only a couple of months earlier. It is not easy, but it had to be done, and I forged ahead with it with the support of MN and my family. I didn't want a divorce but I knew he was gone and I had to end it asap.
Bones you sound so much like I was. I couldn't believe that my loving husband could just leave me, I couldn't believe that he would ever leave his daughter who he claimed to adore. I never ever thought that he would cheat on me in a million years. I thought he must be mentally ill or having a mid life crisis. I think it was a MLC, falling for a woman 17 years younger than himself, what else could it be.
You will get through this. You will have your dark days, but if you could have read my threads, you would see how desperate I was 5 years ago, and how much I have come on since then. You are allowed to feel sad, and to grieve, of course you are, but don't waste your time on him.
Carry on sorting out your own life. He may change his mind in a few months, who knows, but by then you probably wouldn't want to know any way as you will be in a different place and see things differently, so carry on doing things for you now.