If you're looking for state park + gulf, Mustang Island SP has campsites. (I don't know if South Padre is higher priority for you, or coastal camping.) Alas, it looks like the most "parking lot" style campsite area I've ever seen at a Texas state park (looking at an aerial photo.)
The National Seashore area on Padre and the county at Port Aransas allow boondocking on some of the beaches themselves. You might not want to do that for a whole week, but you might be able to get some decent separation between you and the next camper!
Il Carriaggio — 1975 Argosy 24 | Il Progetto — 1976 Argosy 28 Center Bath | WBCCI# 15566
He has all of the virtues I dislike and none of the vices I admire. — Sir Winston Churchill