First of all: Check if the cable (i.e pianowire) at the controls is intact and does not have any kinks and bends that prevents the movement.
Second: There is hot water flowing through the heater core all the time. The heater control operates a cable that is attached to a door in the heater box. Closed = No warm air, Open = Warm air blowing throgh the heater core.
If the door does not seal properly or if it does not close all the way, heat will leak pass the door.
I restored my heater box with a kit from Detroit Muscle Technologies. The heat door has foam glued on both sides. The foam also serves as a seal between the door and the plastic box.
The heaterbox is fairly easy to remove.
1. Remove the glove box cardboard.
2. Remove the plug for the wires. On top of the box.
3. Remove the hoses to the heater core.
4. Three nuts in the firewall around the blower motor.
5. The "hook" in the right fresh air door. You can feel the head of the screw with your fingers. Use a screwdriver. Don't loosen the screw all the way. Just push the screw upwards and try to turn the hook to the side.
6. Mark the cables to the controls and remove the push on retainers. You will probably need new retainers. Hard to remove them without destroying them.
With the heaterbox out you can open the cover and check the state of the heat door.