The only operation aboard ships that I'm familiar with in which the crew both "heaved and hauled" was while working at the Downton Pump, with some hauling on the "bell- ropes" while others heaved on the pump handles; see Hugill's discussion of the shanty "Rolling King" in SHANTIES OF THE SEVEN SEAS.

The operations described for "screwing bales of cotton" into the hold sound more to me as if some form of capstan were being used, and that the appropriate effort would be "heaving" but I suppose "bell-ropes" might also have been used. Where were the industrial artists when they were needed?

The baling of the cotton in the cotton press was an entirely different operation, and at a later stage was powered by steam.

Cheerily,
Charley Noble