If I were looking for a nice quiet stretch of beach, I would do this. Head down to Surfside, turn left at the light on Bluewater highway, this puts you headed toward Galveston, but it's a long drive before you get to San Luis Pass. After you get out of town, head down the highway until you get out in the open air. There will be several entrances to the beach, just be careful and don't get stuck.
I don't really know anywhere on Galveston island where you can find much of a quiet spot on the beach.
Frankly I would ask Dorthy, I bet she knows the Surfside through San Luis Pass area.