You may want to check out
Barnwell Mountain Recreational Area. It's out in East Texas (Gilmer), but looks to be something like you're looking for.
Public land in Texas is much harder to come across than what you're used to in Colorado...especially with the type of use you're talking about here. Your best bet is going to be some sort of private park if you want much off-trail access for vehicles.
Another suggestion would be to find an offroad club in San Antonio as they would have the best local knowledge.