I have an 05 F150 with 5.4 and big cooler tow pkg and limited slip 3.55 rear end.
I though most of the Pickups came that way.
I drove one with the 4.6, don't know what gears it had but it was dog.
If it is constantly slipping and locking it will wear out fast.
I would change the rear gears to 3.73 or perhaps 3.90 and correct the speedometer.
Change the trans fluid and filter and see what is in it.
You don't want the trans to say no out in the middle of nowhere, on a mountain or otherwise. Maybe your trans cooler is too small, also or clogged up. It sits up front below, at an angle the radiator.
Mu son towed a car on a trailer, for a friend about 260 miles each way twice. I checked the condition of the fluid and all was good but changed it anyway. It was about 150,000 miles at the time. Mid grade fuel towing that mush should help with power and mileage and lower engin temps or even premium. I don't know th parameters of the engine control computers though. You will want an extra leaaf in the rear spring pack. Talk to the /dealer and then shop for cheaper. You will want anti slip or limited slip in the rear end, if you don't have it. If nothing is damaged the gear change and bigger coolers and you should do fine. Make sure the Antifreeze or coolant is up to date and clean as well as the serpentine belt spinning at an average higher speed. The button is to lock out the last gear or overdrive. Towing 8,000 lbs don't leave it in overdrive except perhaps some verl level stretches.
chances are you have more than 8,000lbs. If you have seven lug wheels you have the heavy duty. The Salesmen are not up on this stuff. Sound like you don't. You need to increase braking. If nothing else chilled slotted rotors and compatible pads. If you can afford it I would go with Baer or other larger disc kid which would mean larger wheels unless your ten has 18 in or larger wheels, and might still need them.