Our Atlantic Coast Route sections 6 and 7 uses State Hwy. A1A a lot. Section 6 goes from Fernandina Beach (near the GA border) south to St. Augustine, and section 7 goes from St. Augustine south to Daytona Beach, and farther south. On both these maps there are a few stretches that show routing using smaller roads to get through populated areas. And on section 7 there is a paved trail (Flagler Beach to Marineland Trail) that parallels A1A for almost 20 miles. This is what we say in our Riding Conditions about State Hwy. A1A: It is a 2-lane beach access road that carries a considerable amount of weekend traffic but little commercial truck traffic. Shoulder widths are variable.
You can check state bicycle resources and get in touch with the bicycle coordinator for Florida. Many states have online resources as well as printed materials. Nearly every state publishes a bicycle map of some sort that they will send out for free and the coordinators often have more information they can distribute for no charge as well. And while the maps aren't as detailed as ours, they generally offer suggested roads for cycling through their state. Here is a link to the contact information for all of the bicycle or bicycle/pedestrian coordinators:
https://www.fhwa.dot.gov/environment/bicycle_pedestrian/state_contacts.cfmYou mentioned Palatka, so you can also look up the Palatka to St. Augustine State Trail. We show it on our Souther Tier Route section 7 map.
You can also google "bicycling in Florida" and also "Cycling A1A Florida." It looks like there are multiple sites for you to visit for information.