Four-wheel drive not working?

Hello, you can retrieve diagnostic trouble codes from the control module by grounding pin number 13 of the data link connector. I will post the trouble code display information so you can see exactly what you are doing. There looks to only be 4 or so codes you can get. But this will help. I'm going through service information right now, so there will be more to come shortly.
Here are the codes that switch buttons will flash, so 4 flashes indicate code 4, etc. I'll post where pin 13 is as well.

It depends if it is just a global OBD 2 scanner or a scan tool that can do full system scans of every module in the vehicle. Under service info, this method of grounding pin 13 is the way they recommend getting codes out of the transfer case control module. Since you replaced almost everything already and I know these are known for the transfer case motor shorting out to the case or just open circuit inside the motor, but I will post the wiring diagrams for the motor so you can unplug it and check the resistance of the motor to see if it has an open circuit in the motor. This vehicle having obd2 you would think there would be access to the TCCM, but this is the first method it mentions.
Were you able to see the control module on the scan tool when checking for codes? It may also depend if the particular scan tool you have communicates on pin 13, so don't. I will post as much information for you as I can find.

Do you know which setup you have? There are two, one has what looks like a 7-wire connector on the transfer case encoded motor connector and the other has a 10-wire connector. The 10 wire one has a "motor lock" 2 wire circuit that I don't see on the 7-wire transfer case encoder motor, so that looks like the difference between the two.

It's possible the control module is offline, possibly not powered up if you don't see it on a full system scan. Looking at this scan tool, it does say it will scan the transmission, but isn't specific about any transfer case modules. I'd try the pin 13 trick and see if you get any codes, it can't hurt.
And if you have the transfer case motor with the 10-wire connector, (the 7 wire looks like an older version), you can check for a 5-volt reference coming from the TCCM, that will tell you if it is powered up and alive. This is something we do with engine computers a lot to check if they are powered up, if the 5-volt reference is missing, then something is not fully powered up.
But with either setup you can check the encoder motor for and open circuit. When you get time let us know which connector you have on the encoder motor, then we will know which diagrams to work with. This is the encoder motor that does all the work. If I remember correctly, the motor will move in two different directions to engage or disengage. So, one the two wires for the motor, one will be ground and the other power, and then it will switch to disengage.

I would think if the switch is flashing there would be a code present. The TCCM runs its own diagnostics and can store codes, the PCM is in charge of the engine and transmission.
These are the wiring diagrams for your transfer case setup, I verified the wiring colors with the OEM ones, and they are correct.

I posted the wiring diagrams in my last post; you will be testing the 5v ref at the encoded motor unit connector. It looks like each wire for the encoder has a 5 volt pull down circuit. Here in the OEM diagram, you can see the 5volts at the TCCM, and see how there is a resistor after each one. And then a side channel with a little arrow, so P, A, B, C. This means the encoder will pull that circuit to ground when it reaches it.

For example, P will read 5 volts by the TCCM and then when the encoder passes by P it will pull it to ground, and the TCCM will read 0 volts on the Channel P. The resistor protects the circuit from become a direct short to ground. This is how all pull down or pull up circuits work.

And then you can unplug the connector and check the motor itself for an open but check the resistance across the pins where the red and black wires would go. If your meter reads OL, the circuit inside is an open circuit. You can also check each pin to the motor case for a short to ground on the case. It shouldn't read anything to the motor case.
Sorry for all that information at once.

One last thing to add, this TCCM has 6 (12v) power feeds and 4 Ground wires (that includes the Transfer case switch), that's a lot to check. The Grounds also meet a Splice packs in two locations.

Here's a good video just to take a look at, he's diagnosing a transfer case motor issue:

https://www.youtube.com/watch?v=SHXNaU4DpHY

Okay. So, you're saying at the transfer case control module you tested the orange wires (power feeds) and got 0 volts? If there's nothing powering up the control module, don't bother putting in the new motor yet, we need to figure out why there's no power going to the control module first.
These orange wires should be hot at all times, key on or off.
You said you heard some clicking as well?
And these fuses have power?
And one other thing, do you have any of those orange wires going directly to the Transfer case motor, or only to the TCCM? There's a few different designs, and aftermarket wiring diagrams are not always correct.

Are you testing the TCCM with the connector(s) unplugged when testing the orange wires for power? And make sure you're properly grounded with whatever you're using for power checks.
I'm not sure how you got 5-volts okay if there's no power feeding the TCCM though.
There has to be at least one power feed for the 5-volts ref to be there.

Okay, if you have orange power wires going to the motor, then we are on the wrong diagrams, that's part of the issue here. I'm pulling up the other ones.
Okay, the orange wire going to the motor is the motor lock circuit, so that's what you're hearing when you put power to it.
Let's verify these are correct, does this look more like what you have?
These should be the correct fuses, still using the atc fuse 20amp
Sorry for the confusion, there's just multiple setups, and they are all wire differently.