Almost everyone would agree that "Haul on the Bowline" is one of the more ancient shanties, given its reference to a "bowline" which Hugill points out ceased to be an important line to haul on after the 1700's.

And certainly "Roll the Old Chariot" is from the early 1800's, dated by the printing of the revival song of the same name.

Cheerily,
Charley Noble