If your husband is in a position to refinance, it should be made a requirement in the decree with a date certain that it should be done or the house gets put on the market to be sold. If his income/credit is not sufficient to refinance, there are special liens that can be put in place that allow you to take over the house (and payments) if he gets in trouble with the mortgage. You would want to contact a local family law attorney.
If the house was purchased during your marriage, then you are responsible for the mortgage as well. Even if the court awards the house to your husband, the mortgage company will not change the terms of the mortgage to make only your husband responsible. The mortgage company wants to keep both people on the mortgage so that if something happens, they have two parties to go after for the money. Basically, it's completely up to your mortgage company as to whether they will agree to refinance the mortgage with only your ex-husband on the note & responsible for any. & all payments. If the mortgage company will not refinance the note on the house, then you should contact an experienced divorce attorney to determine your options.
DISCLAIMER: Any information contained herein is intended for general informational purposes only. The information should not be construed as legal advice and does not create an attorney-client relationship. Any response given does not result in any further obligation to provide an additional response. Seek competent legal counsel for advice on any legal matter.