An Adventure Cycling map does cover the stretch from Ormond Beach north thru Jacksonville.
A number of years ago, a friend and I followed the coast from Fort Lauderdale to St. Augustine as part of a longer tour. We rode on A1A as much as possible, with an occasional jump over to US 1. As I recall, we left Lauderdale in early April.
We had a wonderful trip with some very pleasant cycling. Traffic was there, of course, but not so heavy as in the November to February rush. Also, auto drivers along the coast seem "conditioned" to expect cyclists.
Without local knowledge, you may want to avoid the Miami area. A better starting point might be in Ft. Lauderdale...north of the shipping port. Just grab A1A and head north. Even in Broward and Palm Beach counties, I'd suggest that you stick to the coast road.
Camping options are limited here, and sleeping on the beach is discouraged and/or prohibited.