If it's running all the time it's probably had it anyway, it's more likely to not be cooling properly than the thermstat - easy to tell, if it's runing permanently then it should be far colder than usual, and all frosted up inside. If it is the thermostat, it's usually a pretty simple job to replace it.
Considering a fridge is all about keeping food safe, you really don't want to be messing about with a duff one.
Playing about with my freezer I found a small leak in the inside stainless skin had allowed water to seep into the insulation over the years and then the insulation no longer insulates.
My freezer just kept going all day and night. I used such a timer for a while but the food starts to get thawed so you have to leave it on longer and longer.
It wont hurt anything then to use a timer and when it gets to be on all day then get rid of the fridge