Steve83 gave you the link to the owner's manual, repeated here.
Go to the diesel supplement page 16, where it gives your GCWR of 20,000 lbs. for gooseneck. Weigh your truck, and subtract that weight from the 20,000 lbs. That is the maximum weight of your trailer and cargo. Subtract the weight of the trailer to get the cargo capacity. Weigh your round bale to see what it weighs. Divide the cargo capacity by the weight of one bale to get the number of bales that you can carry. Look for a trailer that can carry that number of bales. Also check the driver's side door pillar to see what is the GVWR of your truck. A gooseneck trailer carries about 25% of its weight on the hitch. For example, a 5 bale trailer with two 6,000 lb axles would carry 12,000 lb on the trailer axles and 4,000 lb on the hitch. A 4 bale trailer with two 4,500 lb axles would carry 9,000 lbs on the trailer axles and 3,000 lb on the hitch.
2001 F350 XLT 4X4 crew cab, dual rear wheels, long bed, black, 7.3l turbo diesel, automatic transmission, 4.10, shift on the fly, running boards, sliding rear window, 38 gal fuel tank, Odyssey extreme 65-PC1750T batteries, Michelin LTX-MS2 rear tires, Sumitomo Encounter front tires, Shell Rotella Ultra elc, 11,500 gvwr, 20,000 gcwr, 7500 gvw, completely stock, 114,000 miles when purchased, now has 250,000 miles, repairs so far: two starters, batteries, cps, brakes, tires, alternator, ball joints, locking hub, wheel hub, AC compressor, serpentine belt, power steering pump, driver's side window, ignition switch, o-ring, Spectra premium radiator, water pump, fan clutch, fan blades, thermostat, thermostat housing, tensioner. Stolen twice so far, but recovered both times with major damage.
2005 Nomad Rampage toyhauler made by Skyline. 11,500lb gvwr.