Typically what happens next is that on only one salary the person who got the home can't afford to make the payments, or (often due to the legal bills from the divorce) they don't have a credit score that is high enough to refinance the loan.
If the court gave you the home in the settlement, then they should have given you the legal paperwork to sell it. If they said that the home should be sold and the proceeds divided, then if they did not indicate who has the legal authority to sell it, you should go back to the court and get them to make that decision. If they are presuming that you will work together to sell it and your ex isn't cooperating, then again, you need to go back to the court and get clarification and perhaps a legal order to get him to cooperate in selling it.
In the state of Missouri, it is a marital rights state, so your ex typically will have to sign the paperwork at closing indicating that he is waiving his rights to the property if you do not have a quit claim deed on file.
Now, since the loan is in his name, if it is defaulted on, then he's the one who is liable, not you, even if your name is on the title as well as his, (you should clarify this with your divorce lawyer), which means if you just walk away from the home, he's the one who is responsible not you.
The last situation I had similar to yours, they were both on the mortgage and the deed. While the husband was supposed to file the quit claim deed on the property after the divorce he didn't, but he didn't do it because he knew his ex wife wanted to keep the home and couldn't refinance it in her name alone at that point in time. By the time she decided to sell, he had become homeless and difficult to find, plus they were no longer in communications. While I was able to locate him through another family member and get a quit claim deed filed, we also got the bank to accept a statement that he was an estranged husband and it allowed her to sell the home without his signature on all the paperwork. That home ended up being sold as a short sale since she was on the title and the mortgage. It affected her credit for 3 years, but she was able to get the home sold and move on with her life.