You may want to check the heater blower resistor, its behind the glove compartment
Chrysler does not use inertia or reset switches.
The light will shut itself off when the failure that the computer detected is repaired.
If the fan runs with the HVAC panel shut off, the blower resistor power module has failed.
On the auto temp system that happens when the blower motor resistor fails. It is behind the glove box.
maybe your blower motor relay is stuck or burned together ???
If you have an automatic temp system, the resistor block has failed. The blower motor resistor block is located behind the glove box.
The most common problem when a car's heater fan doesn't shut off when the car is shut off is that the resistor on the blower motor has gone bad. The fuse that powers the resistor should also be checked.
try wiggling the wires at the blower resistor if don't work then tap on blower motor
Generally automotive blower motors are designed to run on the lowest speed even though the heater is turned off. The purpose is to circulate fresh air into the cab.
You may have a vacuum leak
check the wireing harnesses
The problem is most likely a faulty blower control module.