Dodge Repair Questions? Ask a Mechanic for Answers ASAP
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The first thing you'll want to do is make sure it's in park and the transfer case is engaged in a gear (not neutral) and slide underneath. Find the wiring coming out of the transfer case shift motor on the left side. Make sure it hasn't gotten down into the driveshaft and rubbed through. Sometimes the harness will fall down and rub on the driveshaft until a wire shorts to ground.
If that looks ok then the next thing that would need to be done is have the codes read from the transfer case control module. Since the service four wheel drive light is on and it won't shift there will be stored fault codes to point in the right direction for diagnosis. Your problem could be the switch, module, wiring, internal transfer case, or the shift motor itself. The shift motor would be most common.