Check the compressor in the back, is it cold,or warm and vibrating?

In my sticky above " replacing GE fridg motherboards and evaporator motors, I stated you must replace both the motherboard and evaporator motor whenever you have problems with the evapoator fan motor. Not doing this will most probably burn out the motherboard as GE uses a weak evaporator motor driver that burns out when the evaporator motor fails.
Just check if the evaporator motor is running at full speed.
If ok, you might have a bad compressor. Gauges and amprobe would confirm.

If the compressor is running, the condenser coils should get warm, check and post results. I don't think your compressor is running, and the click noise is probably the compressor overload cutting out.

Condenser coils are not getting warm (at least they're not getting warm in the 15 seconds that it seems to try to run.
Sequence:
0: Plug in, fast clicks, hum increasing in frequency
17 sec: click off (click does seem to come from compressor and not from mother board), hum goes away
139 sec: click on and humming which increases in frequency
156 sec: click off
339 sec: click on and humming
355 sec: click off
543 sec: click on and humming
559 sec: click off
unplug

During all the clicks on and off, there is no heat coming from the condenser coils.

I haven't located the evaporator fan (is it in the freezer?)

Maybe you are right and the compressor is bad and overload is shutting it down, but how does it explain the previous problem where it needed an external fan to cool? Should I try the old mother board? Should I locate the evaporator fan? Could it be a bad temp sensor telling the compressor to shut off?

When i mentioned previously that the compressor was warm and vibrating, it had run all night before I checked it - it was actually more hot than just warm...

Any advice is welcome - i really appreciate your help!