Your factory hitch is rated for 12, 000 pounds to do what? That 12, 000 pounds is towing pressure. Check the tongue weight limits. Most hitches that are factory insalled have a 500 pound tongue weight limit. What is your trailer tongue weight? Then what is the tongue weight marked on your hitch? I would do that first.
Second. You already said you have problems with wind gusts. Why? When I towed my TT with the Reese dual cam behind my F250 I have never had problems in 40 mph cross winds in WY where we have bad wind gusts all the time. JMHO