That actually might work and if it does it will give you a big hint as to the issue.
Assuming the 'fridge is a 'frost free' model as most of them have been for several decades now, there is a little dodad up in its works called a 'defrost timer'. On the less ritzy versions (you know, the ones with the dial marked 1-9 for temp, as opposed to new allfantastic digital touchpad stuff) it usually hides up under the plastic cover where the temp dial is.
What it should be doing is periodically (time differs with mfg and model) turning off the compressor and turning on soome heat tapes intertwined through the coils. This melts the accumulated ice from the coils and the melt drips down into a pan typically located right behind the pretty kickplate dress panel along the bottom front of the unit. If you access this pan and it is dry as Death Valley in a heat wave over a period of a couple of days, it's a good bet that your unit is not defrosting.
What will happen then is that the ice will accumulate around the coils and eventually will reduce the air passages to zero diameter. When this happens the fantype device can no longer move the air past the coils to be cooled and then blown into the refrigerator compartment. And stuff gets warmer and warmer... An alternate possibility that could produce a similar affect is that the fan that is needed to blow the cold air is itself shot.
Two possible faults occur with the defrost timer. Either the clock motor timing mechanism quit clocking, or the contact set that is actuated by the timerthing is compromised - usually from the contacts becoming arced/pitted over the years and unable to conduct current. Either of these failures is rectified by replacing the defrost timer assembly.
Over the course of the 35 or so years we have had the same Frigidare refer, I have had to change the timer twice. It wasn't (on this model anyway) that hard to access or replace. A typical installation has the wires going to the device terminated with faston connectors so all you need to do is not mess up the sequence as you pull the connector off and reconnect it to the new... Even if you have someone else (translate: repairguy) replace it, it is a magnitude of degrees cheaper than buying a new refrig..
Get the model number of your refrig and do a search for 'defrost timer' for that model number (or alternately call whatever passes for an appliance parts place by you and say the secret words - "defrost timer for Frigidare BlahBlah - do you have and how much?")
Interesting. I searched for “Whirlpool Gold Refrigerator Defrost Timer” and found several people on an appliance repair forum discussing refrigerators similar to ours. It would explain the episodic occurrence of our freeze-up problem.
Unfortunately, I don’t know anyone who fixes things like that in real life.