GE refrigerator stopped cooling and making a constant clicking sound

Our 15+ year old GE GSL22JFRE BS has stopped cooling. As of this morning, the freezer was 37 degrees and the main chamber was 45 degrees. It does not run, but makes an audible clicking sound coming from the bottom rear of the fridge. I do not hear a fan running either. I turned the dial off and left it plugged in. Eventually, it made a light chirping sound, so I unplugged it. I plugged it back in a bit later and it still made the clicking sound.

A parts search shows a compressor overload part as a possible fault. Any ideas? I’d rather not drop the money on a brand new fridge at this point.