It's a different type of camping than you may be interested in (beach camping), but
Padre Island National Seashore, as far down island as your vehicle can get you, might yield the solitude you seek. The only people going by would be 4x4s (and, sadly, tanker trucks -- there's inland drilling at the refuge) traveling even further down-island than you.
The primitive campsites at [
http://wildtexas.com/parks/hcsna.php]Hill Country State Natural Area[/url] are pretty nice. You might have the occasional hiker or horseback rider go by, but they won't literally be going through your campsite/area. Wilderness Camp would be my pick at HCSNA, but there are a couple I have not laid eyes on that could be better.
Lost Pines/
Bastrop State Park has primitive campsites but having not camped in them or even seen them, I don't know if they take you away from the usual foot traffic of hikers.
No TPWD State Parks or wildlife management areas are going to permit ground fires -- too much risk, especially with summer approaching, of Joe Camper setting the whole place up in smoke.