The evaporator coils will always have some frost on them even if only a small amount except at the end of the defrost cycle..
How much frost/ice depends on the amount of humidity in the air and how often and for how long the doors are left open.
Normally a defrost cycle should be long enough to melt all the accumulated frost from the coils. The meltwater should drain from the compartment down the drain tube leading from under the evaporator unit to the evaporator pan under the compartments.
If the drain is partially blocked the water can't all drain away and will refreeze when the refrigerator compressor is restarted at the end of the defrost cycle to cool the compartments back down again to their set temperatures. This builds up the amount of ice left on the evaporator unit each time after the defrost cycle has occurred.
There is a J-curve piece on the bottom of the drain tube, above the evaporator pan and sometimes it gets clogged there preventing all the meltwater from draining into the pan. Check that it is clear and then pour a measured amount of water down the drain tube under the evap unit and check that it all gets to the evap pan.
I'm not sure about your model but in most refrigerators the defrost cycle occurs once every ~10 hours and lasts for about 20-25 minutes. During this time both the compressor and evaporator fan are switched off and the defrost heater is turned on to melt any accumulated ice build up. When the temp in the freezer reaches the defrost thermostat's set operating temp the cycle is ended and the compressor and evap fan are restarted to cool things back down again.
Check that the defrost cycle is occurring regularly.
Also check that the door seals are providing a good seal. Place a sheet of paper between the door and the door jamb and with the door closed, try pulling it out. It will come out with a bit of effort but it shouldn't come out too easily and it should definitely not fall out by itself. Do this at various places around both doors, top, bottom and both sides.
Is there also an ice build up around the dispenser door? If so, the dispenser door assembly may have become disconnected from the operating lever, allowing warmer outside air into the freezer compartment creating more frost/ice on the evaporator unit than there should be.
The mini manual for the refrigerator is GE part #31-51801 which may also help. This should tell you the thermostats resistance values at the various temperatures so that they can be checked. I can't find a free download online but have read that with GE refrigerators the manual can sometimes be found behind the kickplate (toe plate?) or grill at the front of the cabinet.