There are a fairly large number of places you can beach camp in Texas. As mentioned above "north beach" is one offering about 8 miles of camping and south beach ALSO offers another 60 or so miles of open beach camping but beyond the first five miles a 4X4 is required. Another is located on South Padre Island (instead of Padre Island) and across the inlet on Boca Chico beach (South of South Padre Island and directly south of the town of Padre Island) and this one usually means isolation from other campers. Another semi developed area on North Padre is Bird Island Basin which is on the lagoon side of the island instead of the bay side.
Good luck / Skip