The cable-outer moving with the inner is very common and is exactly what causes the flaps to go out of adjustment and make the heating inefficient.
Lubricate the inner and make sure the outer is securly clamped, that should make it work correctly. If you get someone to operate the levers and you get your head down to the dashboard you will be able to hear when a flap is fully closed (closes with a nice thump!), adjust the cable until all is well and you should be there. You have the dashboard vents and also the air inlet flap behind the heater itself, the latter is the most important because if you are not shutting off the cold-air then the cab will never get hot no-matter how hot the heater-matrix is.