On defrost, it is running the A/C (to dry the air), so you're correct on that point. Just usually there's enough heat available to overcome the cooling effect. Sounds like you may have either a clogged heater core (a lot of labor to fix on most cars), or hopefully just a failed actuator (not sure if the Taurus has a mechanical linkage or electronic).
You're probably looking at a good deal of working back behind the dash either way as that's where those parts are usually located. Very sensitive area too, easy to do accidental damage to wiring, etc while in there.