Ive towed 5k with the same truck, 5.0 96 model. Time is the problem. 30 minutes hauling 5k is no big deal- 2 hours is pushing it hard. You need to change all the fluid (either flush or some other method) and get a bigger cooler.
5k lbs is alot of weight behind a 5 liter explorer.
The tranny slowly makes more heat than it can shed through the cooler... each pass of the coolant through the system gets a degree or so hotter every cycle... after 30 mins- 1 hour the tranny cooling system becomes overloaded and it cant shed heat as fast as the transmission makes it. A good big plate and fin style cooler helps alot. I added a giant B&M plate/fin cooler to mine after the stock cooler, switched to synthetic, and never saw over 200 degrees trans temps pulling 5k lbs in 3rd NOT od. It would cruise with that load in the summer here at 160.
Honestly, i think your tranny expert is trying to dupe you into burning your **** up so you will become a customer. Just IMHO.