Padre Island National Seashore has 60 miles of open beach camping. Of course, 55+ miles of that requires 4X4. We like to camp 55-58 miles down, but it is really camping; no facilities, but usually no one within sight of your campsite. You access it SE of Corpus; you can't get there from S. Padre unless you're in a boat.
2008 Rockwood 8272S
2010 Ram 2500 Lone Star CTD 4X4 Megacab
DH+DW 32+ years
Molly the flat-coated retriever
Millie the fawn brindle greyhound
It really depends on conditions, but if you want to camp right on the beach the first five miles is open to 2WD and usually at least the first two miles has good packed sand. Info on camping at Padre Island National Seashore is at Padre Island