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If you walk around the back of the freezer, do hear the compressor running? If not, what do you hear?
Also, if you feel around the edges of the freezer where the door gasket would seal is it warm?
Did you check for the warmth around the front edges of the freezer? It is sounding like the defrost timer is working overtime.
If you get dry ice, you can keep the food that you have removed in another closed cooler.
The description sounds like the tube that allows excessive moisture to drain out (after the unit has defrosted) is clogged causing a back up of water and re-freezing to ice.
If you clean everything out and defrost the back area out then the unit will run correctly initially. You likely will have the problem come back.
If there is a lot of dust and dirt on the coils below, you should clean them off. This could be adding to the problem.
Excessive opening and closing the freezer door will add to it.
A bad defrost timer can be the culprit or a switch.
How old is the freezer? If it is still under Manufacturers warranty you do want to have them become involved.
If you check the pan under the freezer it probably has no water in it because the tube has frozen up.
You can remove the panel inside to access the tube (if you don't have the Mfg. warranty). This will allow you to clear the ice and any obstruction that may be in the tube. You do that with a hot water and a turkey baster. Squirt the hot water in to thaw the ice and remove. You continue this until the water goes all the way thru to the collection pan under the freezer.
Let me know if you would like more assistance.