There is no such thing as a car that shuts off flow to the heater core. If they made a system like that, when you turned the dial from hot to cold, you'd have a very long delay before the temperature changed. Furthermore, it would encourage corrosion and gunk deposits in the heater core, not a good idea!
The system works by ducting the air around the heater core when the dial is set to cold, and ducting it through the heater core when it's set to hot.
Is there any change in temp when you go from hot to cold? Maybe the motor that redirects the air is not working correctly, leaving the heat on.
