From personal experience, I was not allowed to remove his name from the mortgage without refinancing. However, when I went to sell the house a few years later, he signed off on a quit claim deed, provided by the title company. Some lenders may be different, but this was my personal experience.
You should have had an attorney in the divorce process. Presuming you did, ask them about the situation. Don't presume you're going to have to refinance until you talk to an attorney. I've seen homes in Texas with simple quit claim deeds filed that have the result you're looking for. If you have a mortgage, the mortgage company may require you alone to be requalified, but again, get an attorney to get you through this process. If you need references for one, we have experience with 3 very good ones here in Austin, but I don't know where in Texas you are. Best bet is to use your divorce attorney to help you with this.