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Parts for Kenmore Refrigerator 10664934401 - AppliancePartsPros.com

Sears parts have actual part breakdown diagrams. Unforetunately they do not share these with other vendors.

See the attachment for the tech sheet.

You may have two problems.

Is the condenser/compressor fan running?
The reason I ask is it sounds like the compressor is overheating and tripping the thermal protect.

Are the condenser coils and the fan blade clean?

The excessive icing of the vent may be due to a bad door seal.
Check that they are clean.
Check that the hinge side is not grabbing and twisting when you shut the door. if it is usually a light coating of white lithium grease or vaseline will solve this.
Take a two inch wide strip of paper. Place it between the seal and the frame and then pull it out to check the seal. Work you way around both doors every 8 inches or so.

Take a bowl of ice cubes and put them in the freezer. Check them in a days or so. if they are all frozen together then the defrost thermostat may not be opening. This leaves the defrost heater on for too long melting things in the freezer and causing excessive frost.

Lastly check the evaporator/freezer coils. They should have a fairly even coating of ice on them. Be sure the compressor has run for a while because if you check them during or shortly after a defrost cycle then the results will not be representative.
Clogged solid with ice/frost is a defrost problem.
Frost in only one area (usually where the freon enters the coils) is a sealed system problem.