Transmission not shifting into overdrive

Turning overdrive off is a normal condition when pulling a heavy trailer, so that will not cause damage. The place to start is with connecting a scanner that can access the Transmission Computer to see what it is reacting to. In particular, overdrive is typically inhibited until the transmission fluid reaches a specific temperature. If that temperature is incorrectly being reported as too cold, the computer will not allow shifting to overdrive.

Also, look if your truck uses a kick-down cable from the throttle body to the transmission. If it does, those are known for sticking and failing to return under spring tension. That will cause delayed up-shifts at too high a speed, and you may not be reaching the speed needed to overcome that and get into overdrive.