Here's another long shot, Hillsborough State Park. There aren't any sites available the dates you're looking for, but there are individual sites available most of those dates. The problem is that you'd have to move from site to site within the park. Below is the calendar for the time period you're looking for. You can see which sites are open on specific dates. It's not convenient, but it could work for a few days.
http://floridastateparks.reserveamerica.com/campsiteCalendar.do?page=matrix&calarvdate=01/10/2015&contractCode=FL&parkId=281037
