The fan is controlled through that relay, the relay is controlled by three other items, the main computer, a fan timer unit and coolant temp switch "A".
The fan relay (coil portion) also gets its power from the fan timer unit. Two terminals at the relay connector should be 12 volts, check it please with engine running, hot & cold.
How long will the fan run once the key is turned off, I think it should not run more than 15 minutes max. A little more info please.