2003 Mercury Mountaineer actuator door

Hi adreani1;
You just might be lucky and have an easy solution to the your problem.
The constant "Clicking" noise you hear could be your compressor "Recycling".

With the engine running and A/C on, observe the compressors front spindle drive (1) for rotation. You should have Constant un-interrupted rotation.

If it stops rotating briefly and starts again stops and starts, your compressor is "recycling" and is making the "clicking" sound. You need a little charge of freon.

Your evaporator core is located at the firewall inside a split box (1)

If you can change back and forth from heat to A/C your "actuator door" a.k.a. (Blend door actuator) is "not" stuck. If you can not switch back and forth from heat to A/C your blend door actuator could possibly be the problem. I hope it is not the blend door actuator...you will not believe the repair cost for that.
I strongly feel there is no relation to the replacement of the evaporator core in the dashboard and the actuator door switch (Blend door actuator) malfunctioning. The blend door actuator is located right around your glove compartment near the firewall.
I strongly feel you need a little freon that will stop your A/C compressor of "recycling".

If you are curious of what the repair cost would be if it is the blend door actuator...(which I strongly doubt) ...just send me a reply that you want to know and I will give you the news.
Good Luck;
Rick