Hi Shade Tree,
OK I understand. You actually have no hot air getting to where it needs to go.
If both heater core hoses are hot - you have a control issue.
You likely have a seized blend/temp door actuator. Common on Ford/Lincoln/Mercury vehicles since the 90's up. I have replaced many over the years in the Taurus, Sable, Lincoln and so on.
Some Theory: That actuator should react to auto and manual temp inputs from control panel. Problem is, you basically have to remove the entire dashboard to gain access. However, I usually take the dash 'mostly' out. ( I lift it up just enough and support it while I work under it to R&R actuator) Problem is, once you get dash to the point of being able to partially remove, tilt and lift it - the time you save is minimal.
You see - in your vehicle the HVAC/Heater box is mounted so close to the bottom of dash - that you cannot physically get in there to remove the screws and release the actuator.
One Helpful Hint: I also think it's worth putting the Auto Climate Control unit into diag mode. Many times you can get codes which helps make sure you're going in the right direction. Let's say an HVAC Temp Sensor error code was stored - and no other. I'd replace that first, clear codes and retest before condemning the Actuator. But yes, they do fail a lot. I'm including photos.
If you accept I will work with you further as needed.
Blend Door Location:
Top (near center) of heater box, under dash, between top of heater box and underside of dash. Pics included. **NOTE** The image that shows two blend door actuators side by side, be it known, the white one is for a MANUAL system and the BLACK is for ATC. Look close at labels. Images are for reference only and actual production part and/or replacement part appearance may vary. Lastly, the image that shows the top of heater box is showing you where the actuator sits - after it's removed. So the actuator is not in that pic. It's location is.