We just replaced a refrigerator, I have kept it going through several problems the last couple of years by replacing a few components and eliminating some functions such as removing the leaking water filter, etc. (The plumbing within the frig used plastic piping with plastic fittings that failed that I had to replaced with brass fittings) The one problem that had symptoms similar to yours was caused by the defrost timer. The frig was running for the most part non stop in addition to icing up behind panels in the freezer causing inconsistent temperatures so through research and diagnostics determined that the defrost timer wasn't allowing the frig to defrost (every 7.5 hours) and run efficiently. This timer was located behind the toe kick and every time it cycled the mechanical switch made a clicking noise but it became noticeable because it became the switch that turned the frig on and off as opposed to the thermostat and starter relay. You would hear it click then the frig would come on or go off and so like Pavlov's dog every time I heard the click I listened for the compressor to cycle on/off. I installed the timer switch which corrected the icing problem but the frig continued to run pretty much non stop which lead to the real problem all along, the compressor was just about shot and the defrost timer had worn out(probably from acting as the start cycle relay all the time). It was a 14 year old Frigidaire Gallery that should have lasted longer (although the dealer said 13 years was sadly about the life span anymore) and yes I could have replaced/repaired the compressor but it just got to the point to say enough is enough and decided to get a new one. (The new one is extremely efficient but we are fighting with the water/ice in the door feature that is not quite up to spec but with a little tweaking...).
