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Parts for Amana SZD25NL / P1162403W L - AppliancePartsPros.com
The 2 items I'm thinking might be the problem I can't find any info on how to test them.
1. Freezer temp thermostat
2. Compressor running
You said that the unit is running continuously.
I am assuming that you mean the compressor.
You can usually hear it but should also be able to feel it vibrating.
If the above is true then:
The thermostat is OK as it is running the compressor trying to lower the temperature to the set point temperature.
The compressor or sealed system is the problem. It is running but for some reason cannot lower the temperature to the set point.
I am assuming that there is little or no frost on the evaporator/freezer coils since you said the defrost heater works and did not say that they were clogged solid with ice/frost.
A sealed system problem will be an expensive reapir as you need a pro.
Often it makes more sense to put those dollars towards a new fridge.