If you like freshwater fishing, look at Martin Dies State Park in east TX. Many campsites are right on the water. Usually not a busy park, unless a large pack of scouts descend on the place.
If you like saltwater fishing, look at Goose Island S.P., where again many sites are on the water. Also a short drive to Rockport, so you can go out to eat if you like. No beach to speak of, however.
If you like hiking, check out Palo Duro S.P. just south of Amarillo. This park is considered to be the crown jewel of the TX park system. Definitely check weather and park website announcements before you go.
If you like the beach in general, look at Sea Rim S.P., just past Sabine Pass. Only about a dozen or so campsites, so once the day users are gone, there's not a lot of people at all. No shade at the campsites, and no showers either. Bring your own.
hth,
C