Thanks, Claire, I would be really interested to hear that. One of my main interests would be trying to determine if the minor version had been handed down strictly orally, or if at some point along the chain it was a work-up of one of the printed versions.

The tune is an "odd take on the chantey form" indeed -- "Shenandoah" is the only other chantey that comes immediately to mind as something so slow and lacking regular rhythm as to seem tricky to work as a chantey -- my rationalization for that has always been that it was just for really slow work (e.g. breaking the anchor loose) which can't really be done to any regular beat. However, the historical sources do confirm that it was "always" sung very slowly and, judging by the irregularities of transcription, in a very drawn-out and irregular way.
The menhaden chanteys, which are real slow and drawn-out (the nature of the work was different) also provide a model for imagining this sort of chantey. I imagine that if they originated with cotton-screwing, then songs like "Lowlands" could have been the same.