Sorry it took so long to get back, just got back from vacation. To answer your last questions:
How long has this been doing this? Since July 5 we noticed the ice maker was not making much ice. When I checked the freezer temp. I noticed the temp was around 20 degrees. That’s when I set the freezer control a little colder. The freezer responded to that but only got down to 10 degrees. That’s when I cleaned the coils on the bottom of the fridge. That did not seem to help any. I noticed the compressor wasn’t running and that’s when I found the defrost timer and moving the red knob that kicked on the compressor. During that day, every time we checked, the compressor it was running (don’t know if it ever went into defrost mode), this would have been July 7 now. The next morning the freezer temp was around 15 degrees and the compressor was working. I turned the freezer control a little colder (between 6 and 7 on the control) and the compressor kicked off. It took about 20 minutes to pass before I could get the compressor to kick back on with the defrost timer (don’t know if that was coincidental). In the 20 minutes the compressor was off, the freezer temp went from 15 to 30 degrees. From that point on, the freezer temp ranged from 15 to about 25 degrees. On July 9, we emptied the freezer and refrigerator and went on vacation. We came back on July 13 to find the freezer temp below 0 degrees. After checking the temperature a few times, the freezer went up to 15-20 degrees. I replaced the defrost timer last night and this morning the freezer temp was at 10 degrees. But after opening the freezer and fridge doors a few times the temp rose to 25 degrees. As a note, when the freezer is around 10 degrees, the fridge temp is around 33 degrees. When the freezer temp gets to 25-30 degrees, the fridge temp gets up to 40-42 degrees. I do know now that the compressor is cycling on and off consistently.
Is the problem the compressor will not come back on or the freezer temp too high? The freezer temp is too high. After coming back from vacation, it seems the freezer can get cold enough when no one is opening the doors, but as soon as the doors open a few times, the temp climbs. Also I noticed when the unit is not running, the temp rises rather quickly, from 15 degrees to 20-25 degrees in a matter of 20 minutes. Seems like it should maintain temp better than that.
I did a continuity check on the old and new defrost timers and they both performed the same (no continuity between connectors 1 and 3, and continuity between 1 and 4, and 1 and 2 when the timer knob was turned).
Not sure why the unit cannot maintain the freezer around the 0 degree mark anymore. Seems like it cannot keep up with the demand. I removed the rear panel of the freezer last night and noted the top half of the evaporator coils frost up. Reading up on the material you provided indicates there is a refrigerant leak. Do you think it is worth getting this fixed on a 10 yr old fridge, considering I still have an extended warranty on the compressor? Where is the most common place for the refrigerant to leak out from on an Amana?.