I will admit up front I am no expert but fishing in and around Apalachicola is excellent. I live not too far away in Tallahassee and drive through there fairly often.
Going from east to west, some places to keep in mind as possible launch sites as well as beach/pier/dock fishing
St. George Island, FL.
Cape San Blas, FL.
Port St. Joe, FL.
Mexico Beach, FL.
Also there are several rivers such as the Apalachicola River and the Chipola River not to mention several smaller rivers, tributaries, and streams in the area. Locals can give you directions.
Beach fishing is fun and can be productive at places like St. George Island, Cape San Blas, and Mexico Beach (although I stink at it).
And you are not all that far from Panama City Beach which has some excellent piers and beaches from which to fish.
Hope this helps.