Are pirates on Davy Jones's ship dead or partially dead or not dead at all?

The Flying Dutchman (Davy's ship, in the movie) is a "real" ghost ship in historical myth. The crew of the Flying Dutchman are ghosts of men who died at sea, generally after having committed terrible crimes, and they are condemned to serve as part of the ship's crew until their sentence is complete.

The only major change Disney made was to insert the character of Davy Jones as the captain of the Flying Dutchman, and all the in-movie mythology that was associated with him, including the bit about him only setting foot on land once every ten years, which apparently doesn't apply to everyone on the ship. The term "Davy Jones's locker" is another real nautical superstition, although the origins of the name "Davy Jones" are not clearly known, so this character as presented in the movie is mostly made up by the writers.

Barbossa's crew in the original movie, by comparison, were subjected to a curse that made them undead, but they never properly "died" beforehand. Everything supernatural about them was directly related to the cursed gold, and ended when the curse was lifted and they returned to being mortal men, as we see in the movie.

So, they were both undead in their own way, but they were different kinds of undead. Also worth noting, since the crew of the Flying Dutchman are properly dead, the only place they can go if/when they are released from service is onward to the afterlife (whatever that may be.)

Unlike the pirates on Barbossa's ship, The Black Pearl, the pirates aboard Davy Jones' The Flying Dutchman are cursed (or in better terms, bound in service, as it's not technically a curse like the Aztec Gold Curse), rather than undead. They can die, and we see many examples of them dying throughout the second and third movie - although they seem resistant to many attacks that would incapacitate or kill a regular person.

When a sailor dies at sea, Davy Jones will visit them and ask them:

DAVY JONES
Do you fear death? Do you fear that dark abyss? All your deeds laid bare. All your sins punished. I can offer you... an escape-uh. [...]
Life is cruel. Why should the afterlife be any different? I offer you a choice. Join my crew, and postpone the judgment. One hundred years before the mast. Will ye serve?

If the sailor agrees, Davy Jones will offer them a chance - to sail the seas for a fixed number of years (one hundred years) in return to their sworn allegiance to the Dutchman and its master.

A sailor must be dead or dying to receive this offer - they cannot be alive, as evidenced by Davy Jones' accusation of Will Turner:

DAVY JONES You are neither dead nor dying. What is your purpose here?

At the end of this service, they will pass onto 'the afterlife', and face judgement for their sins (perhaps at the hand of God) or more than likely - if they've gambled with their years of service aboard the ship - they will continue serving indefinitely.

It's worth noting that the concept of 'eternity' exists as well, as Bootstrap Bill Turner wagers an eternity of service in a game of dice against Davy Jones.

DAVY JONES [...] Bootstrap Bill, you're a liar and you will spend an eternity on this ship!