I'd start with reading codes, you'll need a scanner that can see the 4WD module. A fairly high end scanner will also show you what range is selected for the transfer case. Without this you have to look, listen and test with a multimeter. There's no sound in the transfer case shift motor for A4WD, you can hear the electromagnet energize with a stethoscope though. You can you hear the shift motor and the transfer case as it shifts to 4H and 4L. Just turn the ignition to on without starting it. Put the transmission in neutral and listen as you turn the 4WD dial.
If the traction control is working the button may simply be disconnected or failed. I doubt the button is needed for 4WD, but the traction control is. The 4WD system in these trucks requires input from the transmission, wheel speed sensors, PCM and ABS.
A4WD has no light, it locks the front integrated wheel ends(4wd hubs) by removing vacuum pressure from them. A4WD also energizes an electromagnet in the transfer case to engage the clutch pack that engages the chain for the front output.
4H locks the front hubs and shifts the transfer case from 2WD to 4WD.
4LO locks the front hubs and shifts the transfer case from 4H to 4L, but requires that you put the transmission in neutral.
You should use 4WD every once in a while so things don't get stuck. Just a few feet straight ahead in 4H and 4L is adequate. I engage A4WD about every 2 weeks just between stop lights in traffic to keep things moving and lubricated.
This thread has some useful info:
Hey fellow ford owners, I've got a puzzling 4x4 issue on my 2005 Expedition that I just picked up. Here's what's happening: - 4x4 lights work during startup check - When I engage 4x4, no lights turn on - Auto 4x4 seems to work (message center confirms, tested in sand) - 4HI light doesn't turn...
www.expeditionforum.com
This thread has more good info on the transfer case including an exploded view diagram on page 3 and on page 4 lots of disassembly pics that might help you understand how it works:
Thanks SafariGoneWrong, that was interesting and a bit different than my AllData info. I thought the clutch was only used in AWD, it looks like from that the clutch is engaged intermittently in AWD and basically fully/constantly engaged in 4H and 4L. I've only done a bit of troubleshooting...
www.expeditionforum.com