KCRG (Craig) is about a 40 minute drive from Fernandina/Amelia Island. 2 friendly FBOs, rental cars (in normal times anyway). Uber from there to Amelia might be reasonable. Fuel prices at KCRG reasonable, but plenty of airports around with better prices.
There's also a public grass strip airport about the same distance away (Hilliard Airpark). I'm not very familiar with what services they might have there, but according to AirNav, fuel prices look pretty good. (https://www.airnav.com/airport/01J
If you're looking for nice beach areas in NE FL/SE GA also take a look at KSSI, Jekyll Island, and St. Augustine. As Butch said, lots of options.