I just spent the weekend at Colorado Bend and I was surprised to find this there:
It's got some pretty good trails and some neat sites. Almost everyone at the park camps by the river, so as soon as you leave that area you're almost completely by yourself.
I'll touch on a few other places. Big Bend is by far the best backpacking in Texas, followed close behind by Guadalupe Mountains. The nice thing about both of these national parks is that they have so much to offer in terms of trails and variety of terrain. Unfortunately they're both 6-8 hours from Austin.
Enchanted Rock, Lost Maples, Pedernales Falls, Colorado Bend, and Bastop are all good overnight or short weekend trips for nearby Austin.
Palo Duro Canyon is great for the panhandle region.
East Texas has a few national forest properties, but the only one I've been to was the Davy Crocket NF. It was pretty good, and no one else was there.
Moral of the story is pretty much the same though: there are some great places, you just have to spend a little more time looking for them.