C.
We went to Galveston last summer. The kids loved it, but we didn't like it. The beach was covered in seaweed and smelled horribly! That aside, we stayed at the Hilton on the beach. It is next door to the Rainforest Cafe. We did rent the bikes that fit the whole family and rode up and down the boardwalk. There is a Ben & Jerry's shop along the way too.
One place we went to last summer was the Hyatt Hill Country Resort in San Antonio. It is close to Sea World. They have a lazy river that connects the pool areas. There is a sandy beach in at least one part too. The kids had a blast! There is enough to do at the resort that you wouldn't have to leave the resort if you don't want to.
I have heard the Port Aransas area is nice, but I can't speak from personal experience.
Good luck!
C.