simply unplugging the sensor most likely would not engage the relay. If it works similar to most systems then the sensor's resistance will decrease with the temp of the engine until it reaches near 0-ohms. which will ground the relay completing the circuit and engage the fan(s).
Ok in simple terms. if the sensor is a one wire sensor, when you unplug it connect the wire to a good ground, then listen for the relay to click. If it does then the sensor is bad, if not then the relay is most likely to blame. Due to the low cost of each and troubleshooting it is giving you a headache, just replace both.
If you do want to continue with the troubleshooting let me know and I can walk you through a few checks.
Here are a few more things to check:
- the relay should have a constance 12V on on of the terminals, if not it may just be a fuse
- is the coolant level correct, if not the sensor may not be in the water and will not read correctly
- one of the fans is sometimes connected to the A/C switch, does one of the fans come on with the A/C?