This happens fairly often. Especially if you have a reason to keep the current home while you move into or remodel the new home, including when the market has slowed and you want to wait for it to recover before selling your existing home.
But do not wait until you are under contract, there is no advantage to waiting to determine your options. Some things to consider before hand:
What current mortgage(s) do you have on your existing home? If you only have a first mortgage, or no mortgage, then it's pretty straightforward. If you have a second mortgage, it could be a challenge depending on how the second mortgage was setup. Texas laws on second mortgages and home equity loans are very stringent.
Do you know what your home would likely appraise for? Texas laws also dictate the maximum amount of a home equity loan relative to your total appraised value.
You need to know the answers to the above questions so that you know what purchasing power you have for the home your interested in. If you already have a mortgage on your current home, and you like your current lender, consider calling them to tell them what you plan to do and they may come up with a new structure for your total mortgages that might surprise you.
Otherwise, consider getting a reference to a reliable local lender that you can work with, your credit union, or even where you keep your checking and savings. If you have a realtor, ask them for references as well. Be open about what it is you're trying to achieve. I think you'll be surprised that even with all the news about the mortgage industry, you still have some very creative options available to you.