Symptoms don't sound like FSR. Usually the fan is indication of bad FSR--cannot control fan speed. Fan might even run when ignition is off. It could be the blend door actuator.
When there is a problem with the system it should default to only heat from the defrost.

First thing I would do is check coolant level if you haven't already done so. If the coolant level is fine I would run a diagnosis to check for errors that might narrow it down. I wouldn't start buying parts yet. there is more troubleshooting to do.

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X5 4.6 2002 Black Sap, Black interior. 2013 X5M Melbourne Red, Bamboo interior
Dallas