You could have a multiple problems. If the fan starts when the engine starts, there's something wrong in the relay circuit. If it starts miles later, there could be problems with the coolant system—low coolant, degraded coolant, stuck radiator thermostat.
The overheat warning bulb lights when there is no resistance in the coolant temperature sensor circuit. A cold engine creates high resistance in the temp sensor circuit; therefore the warning bulb won't light. As engine temperature increases, resistance in the sensor circuit decreases. First, if the engine gets hot enough, the fan will be activated. After that, if the engine continues to heat up at an increasing rate, resistance in the temp sensor circuit will eventually reach its low point, the temp sensor circuit will be grounded, and the warning bulb will light. It is conceivable that your Brick's coolant temperature sensor or circuit has become defective and doesn't detect when resistance resumes increasing during cooling so the bulb stays lit. It's relatively uncomplicated to check temperature sensor function.