Dang N.A....I had no idea the state parks were already booked. Sheesh, I've been trying to get into Stephen F Austin forever! I've been to Livingston a couple of times, and it's pretty nice. I don't think you'll have much problem getting one there at least 2 - 4 weeks out. A real nice park is Wolf Creek on the other side of Livingston (west side). But it's a TRA park (Trinity River Authority) and in addition to the camping fee, you have to pay a per person fee ($4 I think). But it's a really well kept park and nice camping spaces.
Since you mention the younguns, another one to try for them might be Splashway Waterpark. They have an RV campground there as well. Splashway Waterpark & Campgrounds
See if that will get you started. Let us know if you have any more questions.