Not clear yet on what you have but
IF the 4WD switch wires connected to the GEM then you "had PVH" hubs and someone locked them and removed vacuum hoses
So you don't have Live Axle just locked hubs
4WD lights should Flash if shift motor is not working and GEM is
There is another part with the GEM 4WD system, it would be behind the GEM module closer to firewall, thats the 4WD relay box
It has 2 relays inside that connect to the shift motor, GEM operates the relays to send voltage to the motor, these are the Clicks" you may hear when system is working.
But again if you select 4WD and GEM activates the relays but motor doesn't move the 4WD lights on the dash will Flash.
Since there is no Flashing I would suspect GEM is the issue, not relay box or shift motor.
But GEM modules didn't fail that often, so rare but not unheard of
If you have Live axle not lock/unlock front hubs then you have a 4x4CM, GEM is there but not used for 4WD
4x4CM will be in/behind passenger side kick panel, 2000-2003
Looks like this:
https://wholesale.tasca.com/graphics/ChannelAdvisor/Photos/Ford/Watermarked/6L5Z7E453A/0.jpg
Says 4x4CM on the label
These were known to fail.
2001 to 2005 Ranger or Mazda B-series will work as replacement, not 2006 or up
Ford part number is 1L5Z-7E453-AB, or -AC
Dorman has them as 599-252