NH to FL is not east to west! :h
You need to give more information for more informed answers.
You ask if you need a car. How long are you planning on staying? Are you going to one destination for some time or will you be travelling to different parts of the state? Are you already set up to tow with your rig, or will you need to buy everything involved? If you don't have the proper equipment, you might be dollars ahead to just rent a vehicle for the time you're needing it at your destination. You can probably rent a compact car for a week for just over $200. (Renting for the week is usually a much better deal than a day here and a day there.) Of course if towing a vehicle is something you plan on doing often, the investment will be worth it.
The extra fuel costs from towing may make renting a better option. Again, the important question is how long will you be in FL?
Finally, is your rig capable of towing the extra weight comfortably and are you comfortable with towing something behind your rig?