I stayed at Fort Clinch State Park on Amelia Island on my way to Disney. They have two campgrounds, one next to the beach (the sites aren't on the beach, it's down a path) and one next to the river side of the island. At the river campground, there are a couple sites at the end where you can see the river from the site, but most are a short walk. The river sites have more shade than the beach sites. I will also say that the river side has beautiful sunsets! There are videos on You Tube that give you a tour of what the campsites look like.
I remember in my research, Anastasia State Park in St. Augustine looked like it had a nice campground by the ocean, but I didn't have the chance to stay there.
Gamble Rogers Memorial State Park has sites right at the beach (seems to be one of the only campgrounds where the sites are that close to the beach), but I could never find availability. It seems to book up as soon as dates are released!