I'm presuming we are looking mainly for influences rather than origins.
The sporadic evidence strongly suggests multiple origins and we can only guess anyway. The strongest influences on the main body of material when it was at its height seem to be various channels of African/American occupations. Whereas the cotton screwing won't have been the only channel of influence I still think it would have been one channel.

Once this study is relatively complete another interesting development would be a timeline using specific chanties. I would suggest rather than using the dates of references, use e.g., c1825 as the suggested date of when the chanty was being performed. In order to do this effectively it might be helpful to have Master Titles for those chanties known by a variety of titles. I would be willing to help and make suggestions on this one, having already developed criteria for an English folk song Master Title Index. Chanties, however, are very different to other traditional songs and a different approach is needed. For instance with other types of folk song we identify variants of one song by the text and tune, mostly the text. With chanties the text is almost irrelevant so we would look at refrains, tunes and format.