If you don't need hookups I'd recommend Padre Island National Seashore. There are paved campground spots at Malaquite Beach which is right on the windward side that cost about $8. You'll be about 100 ft from the seashore. If you want the lee side you can camp at Bird Island for about the same price. Thy have hot showers and restrooms at both places. If you really want to get onto the beach to camp you can at no cost. The first 5 miles or so to the south are easily driven - just pick your spot. Beyond that for the next 50 miles or so you'll probably need 4-wheel drive. The beach at Malaquite is closed to vehicles for about 3 miles north and 2 miles south. The vehicle entrance to the beach to the south is on the south side and there's a paved road leading to it. The beach "road" is treated like a Texas highway and there is some oil will truck traffic on it, but not much and they're friendly.
The beaches are pretty clean and sandy. You can find some shells but usually have to go about 7 to 10 miles south to find them. There are usually a lot of folks out there doing surf fishing and sometimes the campground hosts at Malaquite offer instruction.
There's wind surfing on the lee side and there's a rental shop there as well.