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Parts for Whirlpool ET8WTKXKQ01 Refrigerator - AppliancePartsPros.com

See the attachment for the tech sheet.

You said "When the dial was turned up, the fan will run but will still not cool."
I do not know which fan you are referring to. There are twa fans in the unit.

Also I am assuming that the unit then runs continuously except when it may go into a defrost cycle.

When set to a colder temperature the evaporator/freezer fan, the condenser/compressor fan and the compressor should come on as they are wired in parallel.

You can usually hear the compressor but also should be able to feel it vibrating.

The first thing to do is determine if the compressor is on.

If on it could be a sealed system problem (little or no frost on the evaporator coils) or a defrost problem (coils heavily iced/frosted over).
With a defrost problem you will get some cooling just not enough.

If not on could be a compressor starter problem, a compressor problem or a wiring problem.