I'm assuming from the structure of your request that you will need conventional hook-ups at each location.
There are some state parks worth considering in that loop, but bear in mind that access pressures are formidable. Today's banner on
the state park reservation site says, "We are now making reservations for arrival dates up to: 12-10-2018". Reservations for many parks are required months in advance, especially if the intent is to stay on a weekend. Compounding that is the widespread flood damage in your travel area.
That being the case, here are a few that you might want to consider:
Houston - Galveston Island State Park
Austin - McKinney Falls State Park (the visitor center was destroyed by flooding but the rest is open)
San Antonio - Guadalupe River State Park (one camping portion is closed for repairs but I think the rest is open)
Based on personal experience, I generally don't recommend Stephen F. Austin State Park (too heavily damaged by flooding and not that nice to begin with... right now it's listed as "closed indefinitely"). I would normally recommend Brazos Bend State Park, but I don't know what condition it's in after the Hurricane Harvey flooding. The closure advisory on their website doesn't give much detail.
In the event that you have the capacity to dry camp, that opens up less well-known options. But if you are renting, that's probably not likely.
Good luck to you.