As a general rule you can't ride the shoulders of interstate highways such as I10 east of the Mississippi. US90 is a good bicycling road, mainly because most of the traffic uses the interstate. 90 also has good shoulders. Get on it as best you can and head west.
Heading west from Fernandina Beach to Yulee is no problem but A1A and 301 on to Baldwin are very bad bicycling roads - mostly shoulderless with fast, heavy traffic. They are being improved in stretches but I heartily recommend you avoid that route. Perhaps you could pick up a county map and use back roads or write a Jacksonville bicycle shop or find someone on this forum who lives in the area. I have bicycle toured through the area numerous times but only up and down the coast. I have driven Yulee-Baldwin numerous times and know of a couple of guys who bicycled it. They said it was by far the worst part of their east coast tour - downright harrowing.