If you are staying near the eastern end of Galveston Island, all of the hotels are on north side of Seawall Boulevard and beaches, if any, are on the other side/south side of Seawall Boulevard.

If you have a car and would like to minimize the number of other tourists you have to deal with, drive further west/south for a less crowded beach.

WRT places to stay, there are multiple places on the eastern side of the island as mentioned across Seawall Boulevard that have access to the beach. You could also look at renting a beachside condo or cottage to the west/south as well as a few resorts there as well.

David