I never tried if I can take the airship at the aerodrome?? But where does it take you anyway, wherever you want, or only to other aerodromes? But then you won't need it, as teleporting works pretty good, there are enough teleport stones around Ivalice.

I don't know when you exactly get the/an airship, I already had the opportunity to fly with the Strahl, Balthier's ship, but that was story based. Or do you get an own one later?