The good thing about Texas is that you can drive for five hours and it's like being in a different country. Everything from pine forest to canyons to hills to coastline.
We do almost exclusively state parks here and find them fairly nice and inexpensive compared to some private campgrounds...MUCH better than KOA and the like. What you get varies by size and location but a little research on google will usually get you a good idea of how it looks. Also, check out the park website for pricing and site specifics.
Texas State Parks - TPWD
Our favorites are Tyler, Pedernales, Martin Dies Jr., Huntsville, and Brazos Bend. We haven't really been very far west or north, so I can't say much about those areas. We like Garner, but hate people....went there for a few days this year because my dad was park hosting for a few months. Just too many people for me (around 400 RV sites + 175 car day use limit). Of course, that was the first of summer.
To keep in mind...reserve online. It'll guarantee you a spot, just not a particular spot. If you find an empty spot that you like better, just call the office and let them know and 95% of the time you can change right there. They'll tell you to stop back by the office after you set up. They'll say that check in/out is 2PM for most places, but I have yet to see them turn anyone away for coming early. You can generally get a spot if one is available any time during the day..I usually come in around 9AM. Also, depending on your needs, not all parks have 50A but most do.
If I were you, I'd also look into getting a State Park Pass since you're looking at several weeks. I can't remember off hand, but it's less than $90 for the year. It'll waive the daily entrance fee (anywhere from $2 - $7 per person depending on the park ) and give you half off the second night campsite four times...so if you visit four different parks on your trip, that's another $40-ish there).
Looks like the uploader might have turned the pics sideways, but the trailer pic is Palmetto over by San Antonio, the rocky/hilly one is Pedernales to the west of there by a few hours and the other is Martin Dies east of Houston nearly to Louisiana.