...and the truck drives in tow/haul? It seems whenever the weather is 50 degrees or warmer it will shut off after 15 minutes of driving, but when it's 49 or colder it woun't come out of tow/haul.
It sounds like you have a switch that needs to be replaced inside of the hollow of the tow bar. It may be just a question of old Lubrication"Grease" or it may be a combination of the two.
Check the circuit first at the tow bar insert.