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Parts for Frigidaire FRT18G6JW5 Refrigerator - AppliancePartsPros.com

See the attachment for a better copy of the wiring diagram.

Yes your temperatures are way too cold.

Unplug the unit and one connection at the temperature control thermostat.
Make sure it cannot short to anything else.
Plug the unit in, the compressor should stay off.
If not there is a parallel circuit path around the thermostat's contacts which is keeping the compressor on.
You will have to track that down.

Did you make sure that the end of the capillary tube of the new thermostat was positioned in the same place as the original?
I am assuming that the original thermostat worked correctly in the past.

If yes to the above, remove the thermostat and check across the thermostat's contacts, should be 0 ohms at room temperature. Now put the end of the capillary tube into a bowl of ice after some time the contacts should open.

It would be rare that you would receive a replacement part that has the same problem as the bad part but stranger things have happened.