We have made that route along TX 105 from Beaumont to Navasota several times. We have stayed several times at the North Houston KOA at Montgomery, Texas--near Conroe and near the Lake but not on it. When we first stayed there it was called 'Haven's Landing' and was not a KOA. But it is still where we stay when in that area.
If you prefer State Parks, you could vary your route just a bit to visit Huntsville State Park, or even Lake Livingston. But the KOA I mentioned is right on 105.
For my part, I'd be sure to drop on down to I-10 before getting to Austin. Maybe just my luck, but I had far rather drive through Houston than the times I've towed through Austin.
Another possibility for a state park stopover is Buescher/Bastrop. I put it in that order because we stayed at Buescher just a few weeks ago. It adjoins Bastrop on the south. Both are in the "Lost Pines" area of Texas.
EDIT TO ADD. If you would prefer to avoid San Antonio, you could work your way around south of Austin to get up to 290 W at Johnson City to visit Fredericksburg and/or Kerrville where you would pick up I-10. We have done that. If your trip is in April you would be in the prime bluebonnet country--if they are good this year.