Ford says you can handle a 5th wheel trailer GVWR of 16,000 if your truck is 2WD or 15,700 if it's 4x4. There's a lot of trailers that you can tow with your F250. You'll also be much happier towing a 5er in high winds than your TT.
My trailer's GVWR is 14,000 and 37' overall length. I towed it just fine with my previous 2009 Chevy 2500 that only had a 5th wheel rating of 13,900. I added airbags to lift up the truck's back-end and better shocks; Bilsteins. And to be honest, I don't feel any difference towing the same trailer with my new dually.