I can tell you its not easy. I did this myself 6 years ago. We had to wait till divorce was final. We got a Divorce Decree that he had to do this in 6 months after the divorce was final. He also had to pay a share of the home value to me. It was agreed that he pay 10,000 in 5 years or contingent on the sale of the home. It was decided that amount after we figured, how much I paid down payment, what I put into the mortgage, devalue, increase in property taxes, increase in value of the home and possible default. It was not a set amount it was just something he decided on. I will let you know, IF he does NOT do this, you are basically screwed if he defaults. That is what happened to me, my ex drew up bogus papers and brought them for me to sign. I was a stupid idiot and didnt read through them. He did it cause re-financing would have upped his payments 1000 dollars a month extra, why? cause the dumb butt had horrible credit and thats why we used my collateral to get the loan in the first place dumb dumb dumb... we also had an FHA loan, and in order to get me out of that it was another 600 bucks to arbitration that he would have had to foot, and a penalty on the loan.
So anyway, I was wondering why I was getting odd letters from Well Fargo, and finally called after 4 years. I KNOW I KNOW I was too trusting. They told me I was not off the mortgage, and my ex was in jeopardy of default, they the house would go into foreclosure in 3 months. GOOD THING I called. The butt hadnt, even started paying me for the 10 grand yet either. So long story short, I poked and prodded him until he got on it and fixed it, but it didnt last he defaulted and had to do a quick sale about 2 years later. SO PISSED. We had to sell for less than we paid for the house, I had to be there for closing and I lived 5 hours away. I didnt get any money from the home, and he had a cleaning service come and gut the house of my belongings that he allowed me to store for some time in the attic till I got my house. YEP all that was tossed. To boot we had to cough up 5 grand to pay off the remainder of what didnt get covered in the sale. I should say I had to cause he was broke...
we also departed on amicable terms, and it was great till I got remarried and had kids, and a different life. He went coo coo for coco puffs. That was the end of that, and now I hate him and we are mortal enemies.
BTW: Divorce Decrees are not a free ticket to get your name off anything, the only thing they are good for is litigation later if he doesnt hold up his end of the bargin. Be prepared to pay a lot of money or he has to pay a lot of money to fix this, especially if you both worked and contributed to the Mortgage. Once he re-fi's with only one source of income it will go significantly up
ADDITION: I suggest you BOTH sell this house together. Get out if it now. He doesnt need that big of a home, and you are already looking for another place. IF you can hang on living together till end of divorce and use this time to sell it together, I am sure your both better off this way. Its final and you wont have to worry about what he is or is not doing. I can bet his payments will not lessen, in fact they will go way up with out you on it.