Switch appears to have an illumination (light) issue as you described the 4x4 is engaging.
The switch housing itself doesn't look very disassembly friendly, as if one could take it
apart to determine the cause of the lighting issue. Power would be going into the switch
to activate the engagement of the contacts for the 4x4 actuator at the transfer case, that
part of it is working.
Could be as simple as the incandescent bulb has burnt out on the circuit board face inside the switch
to poor contacts at the bulb to circuit board itself depending on how the bulb is interfaced with the board.
It may be a soldered connection is my assumption due to vibration issues but may be a plug in type bulb.
Either way, it doesn't look like a easy to work on switch, sort of a "one way" plastic welded assembly.
Tedious to disassemble and repair, you may get lucky and carefully take it apart to figure out the bulb issue.
On the other hand it could turn into a disaster and find that it will not go back together resulting in a useless switch.
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