Ok ,but doesn't the fridge get colder the longer it's running with the door closed?
He's right you know. We once went away for a month, and left a partially opened bottle of wine in the fridge. Before closing the wine bottle up I had used one of the those nitrogen injector kits to keep the wine fresh.
When we got home, everything in the fridge was frozen, and the wine bottle was full of liquid nitrogen!! Seriously...
UPDATE: It may be possible to get a new thermostat that reads in a range that is appropriate for wine, vs a regular fridge. We had to do something similar to our bar fridge. Essentially, there is a small white box (the new thermostat) epoxied to the inside of the fridge, with a wire that runs to compressor. The wiring from this new thermostat is wired into the compressor instead of the old thermostat - that has physically remained in place, but is no longer connected. In your case, you just need a thermostat that reads in a warmer range than most fridges normally run at.
Check with an appliance store that works on older equipment. A place that only does warranty repairs won't have the parts, but a place that works on older appliances will probably have the new thermostat you want.