blowing out hot air through all the vents

This is correct. Your vehicle is a little odd. This was Lincoln's attempt to be a little more fancy than needed.

Basically what happens is the air is always blown over the evaporator and heater core. When the air needs to be cold the A/C system is on and the Evaporator is cold and the air that passes through it gets cold and comes into the vehicle.

However, when you want heat, the heater control valve opens and allow hot coolant into the heater core and the air blows over it and heats the air. So if it is hot all the time, I suspect the control valve is stuck open.

https://www.2carpros.com/articles/car-air-conditioner-not-working-or-is-weak

Below is an illustration of this. However, have you confirmed the pressures are okay and the system is not just empty of Freon?

https://www.2carpros.com/articles/re-charge-an-air-conditioner-system

Please run through this and let me know if you have questions.