[QUOTE="Does the air temperature adjust correctly? That is, do you get hot air when the dial is on hot?"[/QUOTE]
It works correctly in terms of hot and cold air coming out of the correct vents.
I believe you are suggesting I check #3 Temperature Door Motor in the above diagram. Is this also called a Heat Blend Door Actuator (ACDELCO 1573627)? The noise seems to come from the area where #3 is shown in the diagram. Of course, it's easy to be fooled by where a sound is coming from when dealing with a box.
The moving around and squeaking noise stops if the fan is turned on. It seems to search, as you suggest, when everything is off but it works correctly when turned on and seems to hold it's door positions and temperature as set by the knobs. It's that "off" position that is the problem I never had previously.
Thinking through it, I guess the control assembly could be sending data when turned off. It could be on the sending or receiving end based on the diagrams. #3 is likely the least expensive to try, assuming I have enough access to change it.