Welcome to the forums...better later than never. You are correct that sway is caused by not enough tongue weight. Adding steel plate, sand bags, or water is not the answer towing with a F150. Adding enough weight you do not need may solve the problem but it's only a bandaid. The only solution short of getting a lawyer and making them take it back is to make them move the axles back...may still need a lawyer. A TH with only 11 percent tongue weight with no toys in it is unacceptable and with toys in it...its going to take you to the scene of the accident.
If the dealer will not pay to have the axles moved and instead of getting a lawyer, take it in loaded up and have them test drive/pull it after you put the 300lbs of sand in the back with the bikes. Make sure everything is well insured. I would follow at a safe distance in your F150. The video will be valuable.