You're confusing GVW (Gross Vehicle Weight) and GCVW (Gross Combined vehicle Weight). I traded my F-250 in almost a year ago so I won't waste your time trying to quote numbers from memory that I'd almost certainly get wrong.
I was towing a 15k max weight FW. If I was very careful how I loaded things and kept it down to about 14.4k I could make it work but it was a real pain. I was ok on GVW, GCVW and pin weight. The constraint I would hit first was the 6,100# rear axle but I could stay under with empty tanks and careful loading, just barely. Did I mention what a PITA it was?
I'm still towing the same FW but with an F-450. Now I could max out the trailer and dump a load of bricks in the bed while staying well under all the limits. No more PITA.