There's a mountain bike camp 7 miles from Dino where you can camp for just a few bucks. http://www.solavaca.com/

Mack, the owner, will have no issues with a tourer. You can drop the name "Truman suggested I call" and he'll be great. He has water and campsites, and a portalet, but you might want to stop in at Dino and hit their shower before heading there.

The Granbury Lake sites are right in town. Plenty to do, but I wouldn't want to sleep there much.