No, it shouldn't.
Normally a frost free fridge is wired as follows:
1. The defrost timer diverts power from the cooling system to the defrost system for about 10 to 20 minutes every day or two.
2. The cooling system consists of the thermostat in series with both the freezer fan and compressor in parallel. So, when the defrost timer is NOT in defrost mode, it sends power to the thermostat, and when the thermostat calls for cooling, it sends power to both the freezer fan and compressor simultaneously. Now, normally the freezer fan and compressor will both go on together and will both go off together.
3. The defrost system consists of the defrost heater wired in series with the defrost thermostat. When the temperature in the area of the evaporator gets too warm, the defrost thermostat opens the circuit, thereby shutting off the defrost heater and protecting the styrofoam parts in that area of the fridge from possibly melting.
So, in your case, the defrost timer is in operating mode and it's sending power to the thermostat. The thermostat is sending power to both the freezer fan and the compressor motor. But, only the compressor motor is running. The freezer fan isn't.
I'd say that the problem is simply that you need a new "evaporator fan" or there's something in the vicinity of the freezer fan blades that's preventing them from turning.
You're sure that the freezer fan is OFF when the compressor is running? They should both go on and go off together. If the fan isn't coming on, but it is getting power, then the problem is with the fan.