No, you do not need a passport to travel within the US.
If you are flying, then you need government-issued photo ID, such as a state ID or driver's license. Starting in October of 2020, that ID will have to be compliant with REAL ID standards. If you do not have a compliant ID, then a passport will be a compliant alternative. But until then, a driver's license is perfectly acceptable.
Texas is currently a REAL ID compliant state, so your ID should be fine.