Actually the clicking indicates it is already broken/dead. The gear system within the actuator has likely broken a shaft mounting point and the gears are no longer making solid contact or possibly the gears have stripped some teeth and slipping hence the clicking noise while the motor is pulsed to open/close/move the door.

There a actually 3 or 4 actuators depending on whether you have dual zone climate control, all share the same part number. One actuator controls the recirculation door (outside/inside air), one for the mode door (air flow from windshield to floor), and the remaining one or two the blend door (hot/cold).

The only door that typically moves when the vehicle is shut off is the recirc door. It is designed to close (inside air) some say to prevent rodents from moving in.

I suspect the recirc door actuator has failed. I prefer oem/mopar for a replacement. I had an aftermarket one that failed within a year; not happy to redo so soon.