Do you have automatic or manual type climate system? Are the lights for the A/C control head flashing?
This sounds like an issue with the blower power module, it is what controls the fan speed. Most cars use a resistor pack to control the fan speed, this is done electronically. My thought is that the blower power module or the A/C control head is the problem.
About the braking connection, these fans are electronically controlled from a blower power module, and that is controlled by the A/C control head. As the vehicle gains speed the blower fan speed is actually reduced slightly by the control module, this is to compensate for the added airflow through the vehicle as the car is moving.