I would start with checking the temp blend door actuator. There is an electric actuator that moves a temp blend door and they are known to fail. Let me know if the vehicle has manual climate control or auto climate control. The Ford procedure to replace it is to remove the dash assembly but there are ways to cheat and remove it without removing the dash board. I have done a few newer model crown vics going through the glove box. The temp blend door actuator is left of the heater core mounted on top of the HVAC(heater core case/assembly)
If the vehicle has auto climate control we can run a self test to see if there are any trouble codes. I try to change temp and put my hand on the actuator to see if it wants to move. I have seen in some rare cases the temp blend doors then selves break. The actuator needs to be removed to check the door and to replace the door the dash needs to come off. Most dealers will have a guy that is good with this stuff and has done a few. A good dealer will split the labor if they fix it the cheating way, full procedure removing the dash is usually 6-8 hrs.
let me know what you find
lets start here
It is a tight fit to get your hand in and a small ratchet and 8mm socket will be needed. To get a good view many times the radio needs to be removed. This year crown vic is older than the few I have done recently but I do not think much has changed. You could try and ask a shop and see if they are aware of this repair and cut some off the price. It is tricky and there is a chance the door is broken.
circled screws for heater core removal, temp actuator is white
Just allow the controller to reset for 40 seconds. With the key on allow the controller to do a sweep, do not hit any buttons for 40 seconds.
let me know if you get you hand on the (old)actuator if you feel it try to move as you go hot to cold first. Sometimes they get weak and you can remove them and keep them hooked up and as you change temp they move very slow. I use a mirror to check if the slot in the top of the door is broken.
the glove box needs to be removed and then the passenger air bag if you have one.
Again not so sure on this year but sometimes air ducts need to be moved or removed.
you can see by the picture that it is really in there.