If the relay activates the fan and the relay is not being activated, then you can jumper the relay on the LOAD side direct to the fan to see if the fan now works when hotwired. You will of course need to know which side of the relay is the LOAD side, maybe there is a picture on the relay molded into the plastic. Once you know if you have a good fan or not, you can decide to buy one or not.
Next is to know what activates the CONTROL side of the relay, maybe a thermistor switch will close, allowing power to the control side of the relay which will in turn close the load side and allow power to the fan.
Now you should know if you have a bad fan, a bad relay or the component that controls the relay such as a switch or thermistor or another unit in the electronics of the system.
If your compressor is working and pumping as you seem to say that it is, the coils containing the Freon that the fans blow on should be warm or even hot to the touch, careful not to burn your hand by touching more than a quick feel if they are very hot. The hotter they are, the less cooling you will get inside the freezer and ref areas.
On a car we would hose down the same area, the condenser, to see if the air-con gets colder, this test will show that heat dissipation is one of the problems, hence your fan not cooling things down, although this may not be the only problem of a bad fan, as we mentioned above. Can you take your unit outside to hose it down without creating an electrical short?? at least to verify you have enough Freon gas to do the freezing if you can get the fan to do the cooling job (substituted by the water hose for now).