I'm new to Texas myself - having moved here in September (I live in Austin, by the way, if you need a contact) and I have been STRUGGLING to find good places to do quick weekend hikes around here. Yes, Big Bend is AMAZING and there's no one there. Yes, there are established sites and zone sites, but no worries - you won't see another soul. It's not a campground, they are individual sites that are far apart. You also won't see any water. That's the bad part, but easily managed.
I've checked out Colorado Bend state park and Lost Maples State Park, both of them Hill country and both rather nice. Not big long trips, but enough to get away and sleep in a tent and get some miles under your belt in two days. I've heard good things about Government Canyon a bit further south closer to San Antonio but I've not been yet.
You'll find the long mileage further west into Big Bend and Guadalupe Mountains. Seriously gorgeous country and if you can do it I highly recommend it.
But other than that, no, Texas doesn't appreciate your tramping on other people's land. They like private property here and they mean it. The one thing I just can't stand about this place, that's for sure.