Ah! Right you are there and it did not even occur to me that I was assuming frozen ice means cold enough. We bought this from a private party, original owners that said no issues for them but it had not been in use for over a year, aside from them going out and starting it a couple times a month to keep things moving.
We will try the thermistor, and then attempt to do the test you mentioned to test the cooling unit. I am not certain on how to wire it to outside 120v power, but my husband might know how to do this to test that.
Thank you so much for your help.