I'd be shocked if there is a 32' TT that weighs 6k. Mine of that length weighs 9500 loaded for my family, and its GVWR (the max it is rated to weigh) is over 11k. It's empty weight is 7900 supposedly, which is a meaningless number since brochure weights are historically underrated, and nobody tows them around empty anyway.
Weight aside, very few 1/2 ton vehicles (such as your F150) will do well handling a trailer box at/above 30'. The amount wind resistance and sideways sway from wind or passing traffic would be a pretty uncomfortable towing experience. Your truck's light duty suspension, tires, brakes, and engine/transmission cooling are not built for that kind of load.