Four-wheel drive not working properly

More than likely the clicking that you are hearing is the encoder trying to engage but it cannot.

So, the likely cause of this is the encoder has failed but we need to do some testing to confirm this.

Let's go to the encoder on the side of the transfer case and check the voltage on the motor wires and see what it is when you are operating the switch.

https://www.2carpros.com/articles/how-to-check-wiring

I am attaching the wiring diagrams below that will help with this.

Please let me know what questions you have, and we can go from there.

Thanks

Okay. Just to confirm you had no voltage at the encoder, but do you still hear the clicking? When you checked the voltage which wires did you check? The highlighted wires below?

If you had no voltage, then the source of the voltage would be the issue which in this case would be the control module.

However, I would use the second diagram and check the voltage at the module when you are changing the switch state. There should be 12 volts on the 4WD switch and then a stepped down voltage as you change the switch status.