Cannot shift out of 2WD drive?

More than likely this is exactly what the issue is. There are a number of different things that can cause the 4WD to not come on but if you press the button and the light responds and tried to engage then the button and indicator work. However, this could be a control module or the encoder motor which is what actually moves the transfer case into 4WD.

What I would do is start checking voltage at the encoder to find out if the motor is getting the voltage to change position when you press the button. Or at least lay on the ground and listen and feel the motor to find out if it is trying to move.

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

Depending on what we find, we will know better what to do. If the motor is getting voltage but not moving then we need to replace it as that is the likely issue. I am attaching the info below on how to remove the motor so that you can locate it and the wiring diagram so that we can check the voltage.

Please let us know what questions you have.

Thanks.