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Fort Morgan peninsula is plenty remote for most folks, but finding a place on the water is not easy.
But this stretch of shore on the backside fits the bill and has some good bay fishing within easy paddle.
https://www.google.com/maps/@30.2313.../data=!3m1!1e3
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Appalachicola, Port St. Joe, & Mexico Beach are all good choices, Mexico Beach has the best beach and it is also the best place to eat. Appalachicola doesn't have much of a beach but it is a good place to eat. Port St. Joe is the largest town of the three. Mexico Beach is my choice but it doesn't have much of a shopping area. All have good fishing.
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Mexico Beach would be my choice. Fishing is good but women generally don't like it. My wife and daughter were bored to death. I loved it.
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Something on St. Joseph Peninsula would be good for fishing (the State Park even has a few cabins for rent but availability can be tough) but it is the picture of remote. Between Port St. Joe and Appilachicola, it is 30 miles to either one for a grocery store or restaurant so you bring EVERYTHING that you will need when you go. The beach there has won Beach of the Year a couple of times and the bayside is good for fishing too. Stargazing is the only nighttime activity for non-fishing folks.
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Mexico Beach or the state park between Port St Joe and Appalachicola. It is remote. I was thru there in late may and even saw a dead black bear on the shoulder of the road. The state park cabins are heavily booked but good fishing there.