It sounds to me like you have a transmission cooler line cracked or leaking. Though it reads full when you check it, the tranny cooler may be draining back into the transmission giving you a false reading. As you are running, the oil pumps back up into the cooler and starves the transmission. The shifting issue you are having is most commonly caused by low fluid level.
The cooler lines will run from the transmission and either tie into the bottom of the radiator or to an aftermarket cooler in the front of the truck. A leak in that area could squirt oil in the area you describe.
Take it to a transmission specialist. They can find the problem in a matter of seconds.