It’s a stepper motor. A servo works on a very different principal. I doubt it’s burned out. That takes quite a bit of heat and you’d have smelled something.
Steppers work based on the timing of the signals on their input wires. The behaviour you’re describing sounds suspiciously like what you would get if the cable connector was loose or one of the wires servered. Check your cabling closely all the way along.
It’s also possible that the output of one of the stepper drivers on the controller has failed. If your motherboard has stepper drivers mounted as separate units, that’s a fairly cheap and easy fix, but most have them integrated into the controller board.
Try this test:
Use the control panel to move each of the steppers (X, Y or Z) as well as the extruder’s stepper. Do they all work in both directions?
Presumably the extruder stepper won’t. Unplug the extruder stepper and plug the cable for the X, Y, or Z stepper (whichever is long enough) into the extruder stepper. So, let’s say you moved the X stepper cable over: on the control panel, try moving the X stepper forward and back. Does the extruder move accordingly? I’m guessing it will which will disprove the “burned out stepper” hypothesis.
It’s a bit of a pain, but if the motor checks out and you can’t visually see a problem with the extruder cable, try to plug the extruder cable into one of the other (X, Y, or Z) stepper motors, whichever one it will reach. Then, from the control panel, try moving the extruder motor. Does the X, Y or Z axis that you plugged the extruder cable into move?
Plug the extruder cable back into the extruder and whatever axis cable you swapped it with back into it’s respective motor.
If the answer is no, it’s a bit more of a pain, but you’ll have to open up the controller housing (you don’t have to take it out) and swap the extruder cable with one of the other stepper motor cables. Now go back to the control panel again and try to move the axis you plugged the extruder cable into. If the extruder moves then the extruder cable is good and the problem is the controller board.
I hope I’m making sense. It’s 3AM.