You obviously have a defrost issue.
The control you refer to is the defrost timer.
The GE's are noted for heater element failures. After FULLY defrosting the freezer, you will need to pull the panel off the back wall of the freezer to expose the evap coil. There you will find the heater assembly. With the unit unplugged, remove a wire from at least one side of the heater (or heaters) and check for continuity. If you have an open cicuit, then the heater is bad. The replacement heaters on GE's usually also include a new defrost limit switch, replace it as well.
If you have continuity on the heaters and the defrost limit control, the replace the timer.