As luck would have it, i just got back from St. George Island, where my family rented a big house on the beach. Apalachicola is okay, not much there really.
As far as a good beach, I went to the beach on St. Joseph Peninsula, which is less than 20 miles from Apalachicola. It was apparently rated #3 beach in the U.S. and it was pretty damn good. TONS and TONS of shells. The specific beach in question is in the St. Joseph Peninsula State park, down near the tip of the peninsula.
It's especially cool because you can stand on the dunes and it's such a thin strip of land that both the ocean and the bay are less than 100feet from you .
If you're into fishing, the old bridge to St. George Island ( now a really really long fishing pier ) produced some good size shark, Seabass, and giant ocean catfish for me too.