There are so many "Bird Farms" here and they usually have transient spaces to do exactly what you are thinking. If it were not for highway noise, you could roll down your window and listen then follow the clacking sounds of the shuffleboard courts. For a little serenity, I would suggest some of the Florida State Parks along the way. They do however restrict you to a 14 day stay in any one month at each park. Check out
http://www.floridastateparks.org/ Staying there you can day trip to close by parks and get a look see for yourself.
The climate changes near FT. Meyers in that it is a few degrees warmer than the Tampa area. North of Tampa is similar to southern Georgia, several degrees colder, particularly in February. I don't think you will have any trouble finding a place to stay. I'm sure some of the others on the site might have something to offer from the perspective of Snowbirds and what they like about certain parks in the area. Also check out
http://www.campgroundreviews.com/ to see some reviews of area parks.