Texas is a poor place for backpacking because we have so little public land.
With that said, there is a lot of decent hiking places in the area you are describing.
The state recently obtained some land to form
Government Canyon State Natural Area I haven't been yet and it is only for day use, but should be a neat place for hiking.
Hill Country State Natural Area has a lot of trails (40 miles) available for hiking, biking, and equestrian.
Pedernales Falls has some decent hiking trails as well. The primitive area is above the river. It is worth the scramble to get to the river on a hot day! The main loop trail goes back to Jones Spring via Wolf Mountain.

Jones Spring
Also, part of the trail is pretty wide as it is also a road for the rangers.

trail at Pedernales
Another option for the DFW area would be the
Lake Mineral Wells trail system. I have no idea how camping works along the trail.
One final option for the Houston area would be
Lake Somerville Trailsystem. It goes around the lake connecting the two parks. Great trail as it stays on state land the entire time, passing through a game management area. Camping is allowed along the trail.