Hi @carol1001 ,
Given that the compressor turns on when the superfreeze button is pressed and that it starts to cool down, this proves that the compressor is working and that the refrigerant is being pumped though the sealed system to cool things down.
When the temperature control is adjusted and the compressor is stopped may mean that the controls “think” that the temperature is OK when it is not, therefore no further cooling is necessary. The fact that the compressor doesn’t restart when it should when the temperature has risen, also points more to a control problem rather than a compressor problem
It may be a simple fix so perhaps contacting a fridge repairer to check the problem will be still cheaper than a new fridge