Hello, my name is XXXXX XXXXX I will help you with your refrigerator. Your description is good and already covers several possible problems. A few questions should narrow the possibilities. Are the black coils under the refrigerator warm? Is there any frost built up on the lower rear inside the freezer? Thanks.
Sounds like the compressor is not running. It is the black round metal sealed motor in the bottom rear behind the cover. It should be almost too hot to touch and be humming. Let me know what you find.
Ok. Is the compressor, (the large black part) running all the time or do you hear a hum and click every 5-10 minutes? Sometimes using a screwdriver with the handle to your ear will help determine if it is really running. If it is running all the time and doing no cooling you have a serious problem. Either a leak or a total restriction of the sealed system. If it is only trying to start off and on it may be the relay.
This is not good news. The problem is either a blockage or leak of the refrigerant gas or the compressor is not pumping, but that is rare. To cost to repair the system will depend where the problem is found but at a minimum it would be $150 and add any parts. If the compressor needs to be replaced it is over $400-500. Check your warranty but most times it is only covered a year and some are 5 years. Sorry for the bad news but Merry Christmas!