Well, after manually defrosting the fridge by unplugging it and leaving it to warm up for a few days, I plugged the fridge in and left a plastic cup of water in the freezer compartment overnight. Normally, that's long enough to freeze the water into a solid block of ice.

But, the water didn't freeze.

And, apart from defrosting itself once every 20 hours or so, the compressor and evaporator fan were running continuously overnight so far as I can tell.

My next step is to take the freezer floor out of the fridge so I can see the evaporator coils and how frost accumulates on them. At this point I'm expecting to see proof of a low freon charge, and am thinking I'm going to be buying a new fridge shortly.

I'll leave the fridge running with the water in a plastic glass in the freezer for another day or two just to be certain that it's not gonna freeze. Then I'll take the freezer floor out and see what's happening at the evaporator coil.