Although I live in San Diego, Austin will always be my home.  So great to hear you have an upcoming trip there.  

A few recommendations. 

Carlsbad Caverns NP is amazing!  https://www.nps.gov/cave/index.htm

Also, Guadalupe Mountains NP is awesome.  https://www.nps.gov/gumo/index.htm

Interesting fact...Guadalupe Peak, which is the highest peak in Texas, is actually a Permian reef, so you can find marine fossils all along the trail.  Pretty amazing!

Both have camping and are great areas to explore.  Looks like the visitor centers are currently closed and the elevators at the caverns, but the parks seem to be open, as of now and great for exploring and camping.  

Hope you have a safe trip.  

At REI, we believe time outside is fundamental to a life well lived.