Lifelong Texan here:
Recommendations should really be determined by when you'll be here.
September - the State Fair of Texas in Dallas.
March - the Houston Rodeo
With that said, few thoughts:
1) You're not spending all 4 weeks in Texas, right? That would be overkill.
2) In Dallas I recommend a visit to the Perot museum. It's a very unique, very high-tech natural science museum.
3) As part of your Dallas visit, go over to Ft Worth, which effectively acts as its twin city. Visit the Kimbell Art Museum and Billy Bobs at night. Billy Bobs is a mix of rodeo shows and squaredancing - it's exactly what you're looking for.
3.5) As a kid, my favorite part of Dallas-FtWorth was Dinosaur Valley State Park. It's another tube-able river park, but with a dino-dig. There's footprints of a Sauroposeidon (a type of brontosaurus dino) with the prints of an Acrocanthosaurus (a t-rex type) next to it that was chasing it. There's not many places in the world where you can see an actual dinosaur murder scene, so that's something.
4) Travel down to Austin and make a side trip to Enchanted Rock park. Austin is fun for a day or two, with 6th St and bats. If you can't attend a pro-football game (Cowboys or Texans) then try to attend a UT one.
4.5) Between Austin & San Antonio we have Texan wine country and a bunch of caverns they do tours of.
5) For San Antonio, don't just see the Alamo, but check out the other colonial missions, particularly Mission San Jose.
6) Not much else along that coast - rather, check out Big Bend park and continue on to El Paso and Las Cruces (just across state lines). I only bring up Las Cruces because they have a neat "old town" that is from the wild west era, and a jail that Billy the Kid escaped from.
One other thing - look up the pow-wow calendar to see if you can catch any during your visit:
http://calendar.powwows.com/events/categories/pow-wows/pow-wows-in-texas/