Up in the Pan Handle of FL there is Top Sail and Grayton State Parks. There is a bike path that goes from Top Sail all the way to Seagrove and maybe beyond. The path is very family friendly. If you are looking to put more miles in, you can ride in the bike lane on 98 heading West to the 293 bridge, go North and hang a right on 20. Take 20 to 331 and go South back to 98.
The only place I have been to in GA is Stone Mountain. The road around the park is bike friendly.
I havenít been in SW Florida but I did fond this when I was thinking about going. https://www.traillink.com/trail/fred...inellas-trail/
There are places to stay that make this trail accessible from the Park.