When you remove the bad actuator, pop it open and inside you will see a combination of plastic gears driven by a small motor, example shown below. One or more of those gears will likely have broken or worn teeth. Try repositioning the broken one(s) 180°. Could be a more viable $0.00 solution.
Another free opportunity would be to run the self check. I'm not 100% that this carries over to the Expedition, but other Fords with similar looking EATC supposedly will disable the blend door if it detects a fault. Unplugging the car battery or the actuator will supposedly reset everything, but diagnosing the EATC will tell you for sure what the problem is:
Press and hold the OFF button on the head unit, then press FLOOR while still holding down OFF. Now quickly release both buttons and press the AUTOMATIC button.
At this point a little line will appear to rotate in the middle of the display. This indicates testing is in progress.
The test can take what seems to be a long while when you're waiting for it! Just let it run and don't worry.
At the end, the display will begin flashing numbers, or it will simply light up everything on the display and halt.
If only "888" appears for the number, and everything else on the display is lit up, then you have passed the advanced self-test.
If there are numbers, write them down as they appear. Here's what they mean:
022, 024 - Blend door actuator shorted
025 - Blend door actuator not responding properly
030 - cabin temp sensor shorted (either wiring or bad sensor)
031 - cabin temp sensor not connected or failed (open circuit)
040, 042 - outside (ambient) sensor shorted (wiring or bad sensor)
041, 043 - outside (ambient) sensor not connected or failed (open circuit)
050, 052 - solar radiation sensor shorted
This is assuming you have the digital, not manual, climate control. Good luck either way.
The photo Hamfisted posted... you do not need to remove the entire interior trim panel to access/swap the actuator. You just need to remove cupholder trim and the actuators 3 bolts, 8mm IIRC. However, to get a tool (like a screwdriver) down into the keyway to move the blend door, that might just require removing the entire interior trim panel. And hopefully that door isn't spring loaded, or on a horizontal design which gravity will just flop back closed, or a verticle design which might change it on acceleration. Not sure if the actuator just moves the door, or if it moves AND holds the door. Which for me would make the $35 seem worth it.