It is possible for the door to not be moving even though the visible arm is moving. Keep in mind that these doors and arms are 24 years old and could be broken inside the case and not moving.
I will not kid you and tell you that removing the evaporator case is an easy task. If you get air flow direction (I.E. defrost, floor, and dash) then that door is functioning properly. If you have a/c and no heat there is a good possibility of the other door not functioning. If this is the case you are in for a long task if you choose to do this yourself. I would have suspected a heater control valve, but you claim there is flow in and out of heater core.
It has been a while since I have been under the dash of a 5th Ave that old. If you can get your hand on the lever and move it by hand you will probably be able to tell if it is conected or not by lightly tugging on it to see of the arm comes out, or moves in a sideways manner. Let me know what you come up with.
Good luck,
Dave