Nothing I can think of within driving distance is really "camping" worthy. Maybe Lake Georgetown, if you can camp on it (I'm not sure).
Garner State Park - Nice, but a bit of a drive for what it is. And be prepared to not be able to pass anyone, no matter how slow they slow down to during the turns, during long parts of the drive out there. When there is a rare passing section, don't be surprised if there's also a cop behind a tree who doesn't give a damn if you're going faster than the speed limit was in an attempt to quickly and prudently pass an old Grandma 🙂
Balcones Canyon Lands Doeskin Ranch - Pretty and close. Somewhat short depending on your hiking preferences.
Madrone at Canyon Lake - More of a mountain bike path imo. I've hiked it a few times and it's pretty ho-hum.