Welcome to Wild Texas.
Off the top of my head, I'd recommend our two national parks(Big Bend NP and Guadalupe Mountains NP) because they give you more trail options. You could go to the national park service website and do extensive research. If you're not going to camp, I'd recommend staying at the Big Bend NP Chisos Basin Lodge. There's a trailhead just across the parking lot.
Of course, there's desert hiking available also and what better time to try it than the dead of winter.
A must see are the canyons that are carved by the Rio Grande River through the park.
I've never been to the Guadalupe Mtns so maybe others can chime in about this park. However, it is close to Carlsbad Caverns NP across the border in New Mexico. It has a very impressive system of caves and worth seeing.