I travel to Austin frequently on business and by choice stay in Bastrop to get out of the hustle of Austin. Be aware that prices skyrocket when the race is in town. or anything else for that fact
We have stayed at the KOA many times and like it. It's near the Hwy so it's a bit noisey.
Bastrop State Park was beautiful but the fires devastated it, it looks like a telephone pole factory now.
Last time we stayed at The Oaks and it's fine but a bit campy.
McKinney Falls is very good (no sewer hookup) and is secure, it's probably the closest to COTA.