To check the relay you could just temporarily switch it with another and check it like you are doing....
I would replace your ECT sensor. This is what is read from the computer to determine if your fans should come on. If the ECT sensor is not detecting proper tempature then fans will not work correctly. Also there is another sender that is for your temp gauge on the dash. Sounds like you need to replace this also if your gauge is not working.... ECT is a VERY important sensor. There are a number of subsystems that use this info, like atx, fans, engine, etc....
Bob
