First we cleaned out all the coils and blew out all dust and pet hair clogging up cooling vents and coils. Next, I removed the capacitor and relay. I shook the relay, and it made a rattling sound and felt like something was loose in there. Ordered a new relay from AppliacePartsPros.com, slapped the old capacitor back on there, plugged them both back into the compressor on the refrigerator. Plugged the frig in and viola, no more clicking, the compressor runs, the frig cools.
My ice maker was previously not working. It must have been a frozen line somewhere because after unplugging the frig for several days and then getting it back up and running again, the ice maker magically works now.
Major wife brownie points have been accumulated. Good luck!