campground near Austin TX

McKinney Falls State Park is very nice.

I agree with the observation about McKinney Falls. But I don't think you can stay for a month in any state park in Texas. Still, it might be worth it to stay there for two weeks then looking for another.

Try Longhorn RV Resort Niederwald Texas - per their website $775 for a month huge long sites,

Full hookups - wifi and cable

Its just south of Austin - we stayed there for a week early this year - nice pool, lake fishing, and very nice recreation area and laundry - park is only a few years old. Plus multiple fenced in off leash dog areas

We spent a week at La Hacienda last spring and really enjoyed the place. Watched a lot of NCAA basketball :D

We do a yearly stay at Pecan Park Riverside RV Park. It's on the San Marcos river in San Marcos, just south of Austin. Our kids live in Austin (we used to), so we go there to kayak on the river and visit them. Very pretty park and the people who run it are great.
Pecan Park Riverside RV Park

X2 for Hacienda. Have stayed in Oak Forest RV Park., it is a nice park. Lone Star is tight. Hacienda is west and Oak Forest is east and not far from downtown.

To the north is Lake Georgetown which has COE campsites.

Cedar Breaks COE on Lake Georgetown just north of Austin is very nice. Spent a week there last March. Easy access to Austin.