The "correct" versions can often be ascertained by comparison with other editors' versions, however in cases where Hugill presents something unique, there is nothing to compare with.
The latter is the case with these two songs, which don't appear as chanties in other sources. The first, "Larry Marr," has an odd meter shift. The solo verses and chorus seem as if they might have once belonged to two different songs, and the notation does not make it clear how to combine them smoothly.
The chorus sounds to me like it may have belonged to a march from the Civil War era, and I wouldn't be surprised if a deep search specifically for it would turn up something. The solo verse parts, however, seem to be a ballad tacked on. I don't think Hugill ever sang this song in performance, so we'll never know for sure how it was all supposed to go together.
What Hugill did sing was a song with the tune and form of the next one, which he calls "The Five Gallon Jar" in his book, but which in performances was called "Larry Marr." This, as he explains, was not a chanty, and his presenting it is by way of showing the probable origins of the lyrical theme of the first song. My gut tells me that part of the tune for this second song was also mis-transcribed. When I try to sing what's been printed, it sounds terrible—and that's not just due to my lack of vocal skill!
Indeed, Hugill did not sing it as it is written in his book. Moreover, in his performances of "Larry Marr," Hugill wedded the tune (sort of) of the second song to the lyrics of the first. Hugill was recorded singing this creation at Mystic Seaport in his final years. The recording has been well circulated, and since the 1990s it appears that this has been adopted as a fairly popular "chanty" within the sea music revival.
**This phrase is a play on the book's subtitle, about "work-songs and songs used as work-songs."