Infection-free sounds like you want something warmer than north Florida. No runny nose IOW which comes with cold weather.
Start with this map of Florida State Parkshttp://floridastateparks.org/findapark/stateparksmap.cfm
and get a map from AAA and locate where the parks are. Don't stay north of Ocala or Orlando because they get freezes. Inland gets colder than the coast.
There is an online reservation system and it gets booked up early because it's the cheapest place to spend a week.
I can suggest Myakka River State Park outside Sarasota FL on the west coast of Florida (south of Tampa). It's a BIG campground with lots of spaces, playground but it's not Disney World, much more of a natural Florida environment. It still gets to 30-40's at night in February though. And during the day you can drive west and spend the day on the beach for the cost of parking.
Start now, the season has started off very busy, lots of people here earlier than last year. Still time to make reservations though. Best of luck.