It could be a cracked/stripped actuator, wiring, or blend door issue.
I would either take it to the dealer (if that's what you usually do), or go to like Autozone or OReilly Auto, and see if they can navigate (with their scan tool) to your HVAC system and read codes, and actuate the actuators (perform self-test). The dealer is better prepared to deal with it, will cost more, but you'll get a quick answer and know the next route to take. If an actuator is broken or says needs to be replaced try that. They're not expensive: $12-$20. Some are easy to get at, others are not and require some minor dash disassembly.
Deeper explanation:
At certain times, like after starting or shutting down, the HVAC system does a recalibration where it moves the dampers till they bottom out (PCM reads higher feedback amps). Then it knows where each door is relative to full closed and full open. Then it can be precise when the system asks it to be 28% open for example. When something goes awry like broken parts, and the doors don't really close all the way, then the system re-tries a few times to recalibrate. That's what's happening with your system. I have a Jeep with a broken blend door and after a while it'll stop recalibrating. There are a lot of helpful Ram blend door and HVAC videos out there too on Y/T.
Hopefully it's someting simple like a damaged actuator that has a cracked plastic gear or broken teeth. That would be the best-case scenario for you. The other possibility is a broken damper or damaged rod end which is a lot more expensive to deal with. But don't jump to conclusions till you get it checked out.
Let us know what it turns out to be.
