Since you're going to probably need to unplug it for an extended period of time to service the unit, it wouldn't hurt to try just unplugging it (you can take advantage of your perma cold freezer by freezing several plastic foot containers full of water and throwing them in a cooler with all your food which you should remove from the fridge whilst thawing to keep everything cold) Theres a chance that something is plugged/frozen up that could explain your problem.
If you fire it back up and it still wont shut off (toss a thermometer in there, it shouldn't go much below below 0F on most freezers) then you likely have a broken thermostat regulator. The good news is that this is one of the easier parts of a fridge to replace (if it were any part of the coolant system (compressor, coils leaking, etc) you'd probably just need to throw the thing out. The thermostat should just be a little electrical relay somewhere in the unit, if you google your specific make and model you should be able to get a youtube video of someone replacing it or at least a service manual (assuming you don't just save those sorts of things, since they gave you one when you bought it!) Using some combination of wrenches and drivers you should be able to access the offending thermostat. It'll have a couple wires running in an out of it that you should just be able to pull off the unit/decouple. At this point you'll be able to hook up a new thermostat/regulator that you've obtained from ebay/sears/the internets and put everything back together. It should function like normal, do the same tests as mentioned above.
Do not leave this thing running non-stop while you wait for the part to come in... take your food to your folks place/gfs/neighbors or eat it or chuck it, leaving that compressor running non-stop will burn up the motor and ruin the fridge. Which is an option, just go get a new one. Did you forget you are
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