The closest listing to Austin is McKinney Falls State Park, about 6 mi away. 16 places to camp sit within 50 miles, 33 within 100, and 54 within 150. Every one carries weather, wildfire proximity and a straight link to the official booking site, and the federal ones show live availability.
Federal campgrounds where the app shows how many sites are open tonight.
A lake, river or beach you can get into from camp.
The listing says pets are welcome. Leash rules are on each one.
An agency document names accessible sites. The source is on each listing.
Distances are straight-line miles from Austin, not drive times. Mountain roads can double them.