The control unit must be a double pole double throw deal -- it must have Cool, Heat, and Off as viable settings. The current control unit is a single pole no-throw -- it does not have Off or Cool settings, the way it's wired, even though the blower is blowing cold air because that is what is in the pipes.
It is Bitchy Witchy Week, and this Witchy is Bitchy enough to start making utterly reasonable demands.