Ironically the
Chocobo Carriage is sort of an
airship, as the carriage actually lacks wheels, instead being suspended from a balloon above that makes it float a few feet off the ground (while being
towed by a chocobo. As all mounts fly, it really
does fit what you're looking for.
For a literal 'traditional' airship mount, there is also the
Lunar Whale airship
from FFIV now available too.
The manacutter gained as part of
Heavensward's main scenario is technically an 'old school' FF style airship too.
The reason 'gondala suspended under a blimp with propellers' style airships aren't litereally mounts is these are the designs used for
actual airships used to travel between the various cities (specifically the
Bronco-class used by Highwind Skyways on their various commercial airship routes, but there are many others like Cid's
Enterprise, the
Invincible and
Invincible II-class airships, and the
Viltgance class (from FFIX). These can all be made in Free Company workshops and used in Exploratory Voyages as well as being used for FC travel to the Diadem). Hence they cannot be used for personal travel, they're restricted for specific free company use only.