There are three independent flapper doors in the HVAC system.
The fresh/recirculate is is the only one that is activated electrically, with the always on motor that so many of us have had gears strip.
The temperature blend door is cable driven mechanical. Some, especially early 03's had a problem with it not being properly adjusted and allowing more heat through than desired at the cold setting. That's an easy fix
viewtopic.php?t=1004 That will not affect air direction issues, only let the temp door fully close when the dial is at the cold setting
The air direction door is cable driven mechanical. Sounds like this is the one that is giving you problems.
Page 3-2 of the 03 owners manual says the airflow should be:
Mode Knob
(Vent): This position directs the airflow though the
upper instrument panel outlets.
(Bi-Level): This position directs the airflow
through the upper instrument panel vents and toward
the floor.
(Floor): This position directs the airflow toward
the floor and through the right and left side vents. The
side vents can be shut off to allow for more air to be
directed toward the floor.
(Defog): This position directs the airflow
toward the floor, the windshield and the side windows.
The air also comes from both vents at the ends of the
instrument panel.
(Defrost): This position directs the airflow to the
windshield and side windows. The air also comes from
both vents at the ends of the instrument panel.
and I only mention that to point out that by design, air comes out of the side dash vents no matter what position the dial is set to. So if that is all you are experiencing, that's normal
One end of the cable is connected to the back of the HVAC controls. Once I tried and failed to remove the HVAC controls, I couldn't figure out how to disconnect the two cables from the back, and I may have broken part of the one of the two cable retainers. Mine still works, but it might come undone someday. Since you bought the car used, and it seems to have always had the problem, perhaps the previous owner had some work done such as trying to remove that to replace a light and whoever did it was as inept as I.
The other end of the cable is attached to the driver's side of the airbox, roughly straight up from the gas pedal. Here's a pic looking up from around the gas pedal, the yellow box shows the cable attached to the plastic arm of the door.
- modecable.jpg (65.12 KiB) Viewed 2085 times
It's possible to reach up there with one hand and use the other hand to rotate the mode selector knob to verify that the moving the knob causes the plastic arm on the door to move. If it doesn't, then there is a problem with the cable, either broken r disconnected. If the knob moved the arm and it has no effect on the airflow, then there's something unpleasant inside the airbox, a disassembly and repair task that I don't envy