I would agree that salt absorbs moisture, its in its nature to do so.
I'm amazed that here on the west coast, or wet coast as many call it,
salt does not seem to cake even at high humidity levels. Even so, I
would think that salt would cause some instabilty in whatever wood you
choose. I'd go for an insert of some description to keep the salt from
being in direct contact with the wood. Or, you could seal the inside
with epoxy, use a water-thin variety and make sure that it is well
sealed then you effectively have a plastic insert in the shaker.

John