by now, you probably have defrosted your "frost-free" refrigerator, and it's working fine.
but the ice accumulation may return and cause the problem once again.

in a refrigerator which has a self-defrosting feature,
no ice is supposed to accumulate on the coils, unless it is melted off on a defrost cycle by heater coils.
it sounds like your defrost timer clock motor died.
some defrost timers are accessible from the lower rear of the back of the refrigerator.
if you can advance the motor with your screwdriver thru its access hole to a cycle which defrosts your icing, then the defrost heaters are ok, so just replace the defrost timer.

not sure? the defrost timer advances like a clock. mark the inner dial and make an opposite mark on the housing. come back in a half hour and see if it moved. if it moved, the timer motor is probably ok.

so get your digital multimeter out and follow the schematic which is hopefully on the back of the refrigerator.

or, check the year of manufacture. old=costs too much to run it compared to the new energy efficient refrigerators.
so cough up the 500 bucks and get a new fridge already. the payments on it will be made by the electrical savings over its 5 to 10 year life, so the new fridge is free to you if the old fridge is inefficient.