BTDT with a 2010 F150. JMO so take it how you want, but that 6sp tranny is the worst tranny for towing I've ever towed with. The reason it's downshifting alot is because you can't lock out either OD's or select what max gear you want. The tranny will hunt ans shift all day long. I had the same issues towing 7300lbs going over simple interstate overpasses. It would shift 3-4 times during the whole process. Really irritating.
the plus side to the tranny is they greatly improved the cooling from 08. I used a Scangage to monitor tranny temps and the tranny stayed in normal temps all the time.
Tow haul mode won't change often the tranny shifts. It will just make the shifts firmer. BTDT