Not sure what your asking for, Galveston Island State Park is reopening some more spots in July. Prior to Hurricane Ike it was a nice park with sites near the water and near the bay. visit the link below for details. At Galveston you were not on the beach per se but less than 50 yards away from it. There will be little if any shade at a state park beach camp. Fishing is a lot of fun though
TPWD: Galveston Island State Park
Other state & national parks with beach camping include Mustand Island and PINS (Padre Island National). there are trip reports on this site both are close to Corpus.
Beach camping itself can be a challenge as its always windy and you get sand EVERYWHERE but you can survive both conditions and have fun.
Let us know what else your want to know