No help on campgrounds from here, wife thinks camping is a motel without room service. Several state parks in the Hill Country are worth checking out. Years ago camped at Inks Lake, very scenic, part of a chain of lakes along the Colorado River. Lots of campgrounds along the rivers, shouldn't have any trouble finding something good.
x2 on camping at Inks Lake State Park. Good shade and water access, and well located for rides and sights (Longhorn Caverns is nearby).
There is also a cool place up on Lake Buchanan if you are interested in nature hikes and star gazing...it’s called Canyon of the Eagles. They offer RV Park, camping, or nice hotel rooms. Relaxing indoor/outdoor bar and great views of the lake from the restaurant.
Several private camping areas on Lake Buchanan as well.