Just had a great weekend at Huntington beach state park here in SC. The primitive section is very hammock friendly but no alcohol allowed.
Capers Island is awesome, but on the weekends you will have a whole lot of people pull their boats up on the beach to hang out. The forested part of the island should be pretty secluded though - they don't venture far from their watercraft. I grew up a couple islands down the coast and we used to go there a lot by boat - never camped though.