You missed Guadalupe River SP
, which is a lot closer to San Antonio since they were looking for something near SA. Guadalupe River is a big park and is reasonably close to New Braunfels, also, with its German culture.
There are parks in Johnson City which give easy access to the LBJ birthplace and LBJ ranch. The big house at the LBJ ranch is now open for tours. Fredericksburg, with German culture, is an easy drive. That area is worth a couple of nights on the way.
Also, at that time of year, I would stay no further north than about Georgetown. I have had frozen hoses overnight in late February in the Austin area. The San Antonio area gets more influence from the Gulf of Mexico and usually the night time lows are a bit less severe.