I had a work buddy with a 2001 with that same symptom. His occured after his son left the dome lights on which drained the battery. He jump-started it a couple times, but the battery had to be replaced. His 4wd would not work (same as yours) after (the jump-starts and) replacing the battery. He replaced his 4x4 Control Module to fix it.
http://www.therangerstation.com/tec...ng_1995_&_Newer_Electric_Shift_Transfer_Cases
Trouble Shooting:
1995 & Newer Transfer Cases:
This problem turned out to be the electronic 4WD Control Module behind the passenger side kick panel. The box is easy to replace and has 4X4CM on it.
Symptoms:
1) When starting vehicle, the “4WD” and “4WD LOW” lights on dash never illuminate.
2) After turning 4WD switch (knob on dash) there is no response at all, 4WD light on dash never illuminates and no clicking.
3) 4WD will not engage.
Normal operation would be:
1) When starting vehicle, the “4WD” and “4WD LOW” lights on dash momentarily illuminate right after starter engages and key is released to the “run” position.
2) After turning 4WD switch (knob on dash), 4WD light on dash illuminates and clicking is heard.
3) 4WD engages.
4) By the way, the “4WD” and “4WD LOW” lights aren’t supposed to illuminate when the key is turned to the point right before the starter engages. They only momentarily illuminate AFTER the truck is started. All the other dash lights light BEFORE starter engagement, but not the 4WD for some reason.
Before changing out the CM, I did this (so I didn’t waste $100 on a box I didn’t need):
1) Checked the electrical grounding lugs near the module with a meter.
2) Checked any fuse that said anything remotely about 4WD or powertrain.
3) Checked ALL fuses I could easily check. TIP: The ones that are smaller in size have 2 very small metal points to check with a meter without needing to remove the fuse, this saved a ton of time.
4) Checked the relay under the hood that said something about powertrain control module or PCM (I changed it with the one next to it for the blower), they seemed to be rated the same from what I could tell. I swapped them and tried the 4WD, no change.
5) I had a diode setting on my meter; I checked the powertrain control module diode.
When installing the CM, I disconnected the negative battery terminal first. Also, I’ve had past problems with another vehicle and the loss of the computer memory for idle control. Granted, I don’t know if this applies to the Ranger, but I played it safe so I didn’t have to redo it. So to “relearn idle” I did this:
1) Park vehicle in a place I could let it run for 10 min without moving.
2) Let engine cool several hours to attain a “cold engine”.
3) Disconnect battery negative terminal.
4) Changed CM.
5) Reconnect battery negative terminal.
6) Started vehicle, and let idle in park for 10 min on cold engine to relearn cold idle. Absolutely DO NOT drive or rev engine at all.
Then I put the 4WD on and drove crazily around the neighborhood on the snowy roads. Fixed! Whoo Hoo! Now I can plow my driveway again and save my back! Super Whoo Hoo!
I got the 4WD Control Module at
www.directfordparts.com for $98 (plus shipping). The local Ford dealer wanted $185. The local dealer also wanted $80 to diagnose the problem. I had no way to test my CM, but took a gamble and it paid off. Whew!
^Pasted text from the link at the top