The HVAC Control Module sits behind the glove box and contains the vacuum valves that open and close the vents that direct the air flow in the cabin, and has the motor that operates the blend door that controls the ratio of cooled air vs heated air. These blend doors tend to gum up a bit and stick if they aren't used much as you seem to be experiencing. The motor that moves the door is very slow and weak as designed and sometimes can't budge a sticky door.
How to test the HVAC Controller motor:
First thing I wound suggest is to pull the glove box and find the HVAC Control Module. At the top you will see the white plastic rod that goes to the blend door, and where it simply clips to the lever arm that is attached to the motor.
Un-clip this rod and move it left to right and feel how much resistance there is especially at both ends. With the rod still disconnected, start the car and turn on the AC. Adjust the temperature setting to command max heat and observe the lever arm and the motor the lever should move to the right. If it doesn't move or when it stops moving, command the system to max cool. The lever should now move to the left.
This will tell you if the motor is working or the blend door is the problem.
If the motor doesn't work, just get a whole HVAC Controller unit and replace it.
If the motor works, you can try to lube the linkage to the blend door as best you can. (It's hard to get at.) and just move the rod from side to side and try to get the action to smooth out some so the motor is able to move it by itself again.
How to get cool air with a bad HVAC Controller
With the head unit still set to max cool and the rod disconnected, move the rod fully to the left and then clip the rod back onto the lever arm. As long as you don't turn the heat back on, you can ride this way all summer, even if you don't fix the issue, until you turn the heat back on.