Well, I wish you'd posted last week instead. We'd have invited you to join us (the Heart of Texas Camping Unit) just a few miles from you, at LBJ National Grasslands near Decatur. It's all boondocking, but it's a great place to camp and we had a fine weekend.
Fort Richardson in Jacksboro (already recommended above) is a nice little park and quite close to you, it would be good for a "shakedown cruise" weekend where you're close enough to home and civilization to easily sort out any problems you might encounter.
The City of Hico has a nice little municipal park with some full-hookup sites (and some water+electric sites) that's close to the town but still quiet, unless you happened to be there on a weekend when our unit was camping there.
Dinosaur Valley State Park by Glen Rose is a nice park.
I've stayed in several of the places recommended by others above as well, and wouldn't argue about those recommendations.
One thing to remember is that while you'll encounter a day or three that are too cold for fun camping (with the occasional ice storm), overall what passes for Winter down here is ideal camping season in Texas! If you're trailer is by your house, you can just plug in and keep the furnace going on the occasional cold night, or you can get good and winterizing and de-winterizing quickly. It's June through most of September when it's miserable to camp around here!