You will pay a premium for a campground in Florida in the winter. $800 a month seems low for Florida in prime season for a place you want to visit. Even at $800 a month you will be paying $4000 to park your RV. For 6 weeks that is $666 a week.
I would estimate you will be closer to $1000 a month (or more) in Florida, which makes your price $5000 for the winter or over $800 a week.
That's not including the cost of driving down and back.
I would suggest you find a couple places you like, plan out your vacations and make reservations for a week at a time. It is MUCH cheaper to store your camper somewhere than to rent out a spot. Every time you leave, I would winterize. It can freeze in Florida.
If it were me, I would take a month long vacation all at the same time and go down ONCE. Spend $1000 +/- for a nice place on the gulf and enjoy your 4 weeks vacation. If you were going to take 5 or 6 weeks vacation... why not clump it all together?
