The cold control seems to be the weak link on GE's from the mid to late 50's. The one on my '57 Combination got stuck and wouldn't shut things down when it reached proper temperature. It was getting down into the lower 30's in the refrigerator section, and the serpentine coil had a thick layer of frost building up. A NOS replacement cold control fixed the problem. Larry at Modern might still have one laying around.
Since I put my Combination into service, I've had a cup of water on the bottom shelf (where the beer goes) with a digital thermometer probe in it. With the bottom shelf averaging 38 degrees, the top shelf near the serpentine evaporator hangs at about 36, which is about as cold as I'd want it. The defrost water in the trough usually freezes into a long bumpy chunk since it's so cold back there just a few inches from the evaporator.