I'm just guessing from looking at the diagram; appears to me that the probe is looking for coolant to conduct a small amount of current to ground, the module is trying to sense that current and turn on the light when the current is missing. So I would probably start by connecting a jumper around the sensor to ground. If the light goes out, likely the sensor itself is bad (or eroded).
If not, try disconnecting the lamp wire from the sensor. If the light still stays on, there is a short or miswiring somewhere else. After that, it's probably the module.