Blend Door Problem
Your problem sounds like the common blend door problem on the Ford Expeditions, and Ford Explorer. The door is on a vertical axis and when it breaks, it can swing back and forth randomly giving heat or AC or nothing. The difference on acceleration is the force placed on the door swinging it one direction.
You can check the operation by fully opening the glove box, removing the actuator motor on top of the plenum box and examining the blend door axle (the actuator is under the unit on top of the hump behind the flexible black plastic cover. It is on top of the unit, behind the glove box in the 2002 F-350 though). Pictures of the common fail signature are available on the Related Links below. There are several "free" fixes that mostly are worth what you pay for them. The solution on the web site is solid and proven over hundreds of vehicles and will resolve the problem once and for all.
90% of the time it is the temp mode door actuator located in the driverside rear of the truck
The 2005 Ford Expedition has 39.8 in. of rear head room.
The 2005 Ford Expedition has 64.3 in. of rear shoulder room.
The 2005 Ford Expedition has 38.7 in. of rear leg room.
The 2005 Ford Expedition has 62.4 in. of rear hip room.
The rear track of the 2005 Ford Expedition is 5 ft. 7.3 in. (67.3 in.).
The 2005 Ford Expedition has rear wheel drive.
change the light bulb,
rear a/c blend door actuator bad [broken plastic gear]
replace front heater corefront and rear have independent heater cores
Possible blockage in the rear air units expansion valve. I just replaced mine for the same reason.
one of the fan belts is broken.