|Three places come to mind. Big Bend National Park, Big Bend State Park and Padre Island National Seashore.
Big Bend National Park has three different campgrounds. We stayed at Chisos Basin which is cooler than the other two campgrounds and tends to fill up the fastest. The campground sites are small and you cannot have a fire.
Big Bend State Park is very nice and VERY remote. Unlike Big Bend National Park, where people are right next to your camp site, the state park has a lot of room. There are two different areas to camp. One area is on the Rio Grand river and my wife and I spent one very nice night there. The other area is way back in a dirt road and has very remote camp sites that are inland. You may be camping miles from the next person. Some of these sites it would be impossible to get an RV into or even a trailer. We took a "2 wheel drive" site with our Tundra truck and actually ripped off the hitch electrical connection getting to our site. But boy oh boy was it nice!
Padre Island you are camping right on the beach. We had a great time, but you better like wind!