I have almost the same problem on my LG LSXS26326S side-by-side refrigerator about 7 years old, and suddenly one day the ice crushing auger will not stop unless the refrigerator is unplugged. I was able to remove the entire ice maker unit, but like you, I don't know how to diagnose if they problem is specifically with the part (motor for auger of the ice making unit) or if I need to replace the control board. As you said, both are a little pricey. I want to learn from your experience. Did changing the control board fix the problem for you?

Hoping for some help identifying what part I need to fix my fridge, its a Fridgidaire FGHS2631PF0 side by side. The ice maker auger motor won't stop running with the door shut. The switch for the freezer door will make it stop when I open the freezer door, but it will start again if I press the switch by hand and the fridge "thinks" the door is shut.

I took the front panel apart and there is one switch that senses the paddle being pressed for either ice or water, and another that senses when the actuator has the "ice flap" (for lack of a better word) open or closed. I tested both switches with a multi meter and both are working. In addition, it seems that if the paddle switch was bad and the fridge "thought" the paddle was pressed, it would dispense water when on water mode, and it doesn't, the water dispenser works as just it should. I'm thinking maybe there's a control board for the ice maker that's bad? I don't want to just throw parts at it until I get the right one. Any help is appreciated, thanks.