If you aren't stuck on saltwater, or being near the redneck riviera, check out Blue Springs Park in Gilchrist county. A bit due east (over the water) from Cape San Blas but beautiful springs, river, camping allowed, I think they allow dogs, etc.
Blue Springs Park, High Springs, Florida - 386.454.1369
They do close for 2 weeks before/during christmas/new years but they are definately open all January and on