GVRW is based on the maximum capacity of the axles, springs, tires and coupler, and is the MAXIMUM SAFE weight of the LOADED trailer, not the weight of the trailer its self. If the actual trailer weight is less than or equal to the max trailer weight of your vehicle, you CAN tow the trailer with a weight distributing hitch.
All that being said, it WILL be a chore. Your problem is going to be stopping being able to control sway. If you're willing to drive 40 or 45 mph MAX, you'll probably be OK.
My best suggestion is to tow the trailer close to home and see how the rig works before you head out to Florida. Having to miss a day or two at a lake 100 miles from home because your rig isn't safe is better than missing a week in Florida, or worse yet, a crash.
I'll bet you wind up with either a different tow vehicle or a smaller trailer.
I tow a 21' with my '02 5.4 L. I have enough power and brakes to make ti go and stop, and with a WDH the sway isn't a problem. I'd be reluctant to regularly tow a load that was much bigger than what I have now and I've been towing trailers for close to 40 years.