Another favorite is The Cloisters at Sea Island GA and although I haven't been there I understand Jekyl Island has a lovely hotel. When I was at the Cloisters it was cash only, no plastic. I do not know if this is still a rule or not. If you have young children they run their family special in July and August. Not cheap, but your kids get to act like adults and I think they are free.
I also like the Ritz on Amelia Island. Lots of fast food places & other restuarants on the Island so you don't have to pay Ritz prices for all your meals. You should have seen what I was able to bring in from the grocery for breakfast. Would eat lunch at the hotel on the beach, and then go off the property for dinner. And some summers they run special rates that are not expensive.