I figured that it was either the thermostat or the defrost timer. I ordered the parts without even opening the rear panel. Well, I should have taken it apart first, as the defrost heater was burned out and in two pieces. Anyway, I then ordered the heater and installed it. Now, everything is working fine.
If you have this problem, I suggest taking off the back panel in the freezer to check things out first. Also, you might use a space heater to manually defrost the evaporator coils. It's pretty quick and then you can start it up again. Mine ran for several days before the refrigerator would start to warm up again.