If you don't mind being 30 to 40 minutes north of Jacksonville try Fort Clinch State Park on Amelia Island.
The state park has sites where the bay meets the ocean. Right across the bay is Cumberland Island where wild horses run free, very undeveloped.
At Amelia Island you may enjoy sunrises and sunsets. Head into Fernandina Beach(on Amelia) look over the Intracoastal Waterway while you dine and watch the sunset.
Ameila Island is very pet friendly if that matters. You may walk your pet on the beach (not in the state parks), take them to many outdoor restaurants, etc.
Before the end of May and after Labor Day the beaches are empty. Not uncommon to not have anyone within 50 yards of you.
There are many places on the island you may drive on the beach! At the South end there is 5 miles of beach to drive on, you *must have* a 4x4 to do this, in this state park pets are not allowed on the beach.
Amelia Island is the sole reason we bought a four wheel drive towed
Here is an example of after labor day on the South end. BTW: It's only 15 minutes to anywhere on the island.
Steve