I have good but somewhat peculiar news. First of all, Gale's quotation is quite accurate. Second, Dana wrote at least three different versions of the same paragraph, each one different, for no obvious reason.

The version that includes "Cheerly men," "Dandy ship and a dandy crew" and "Tally high ho!" comes from Dana's 1869 revision of his 1840 publication. That's a nearly thirty-year difference. Those three titles do not appear in the earlier versions of the paragraph.
The 1869 edition is the one usually reprinted.

Gale cites instead the edition published by John H. Kemble in 1969, which combines the wording of the 1840 first edition with that of Dana's original manuscript. Kemble restores material that had been edited out, presumably by Dana's publisher. According to Kemble, Dana's manuscript paragraph includes the title "Grog Time a Day," just as Gale quotes it. Also significant is that Dana originally wrote "Round the corner, Sally!" rather than just "Round the corner." That makes it more certain that the shanty with the fuller, familiar title is the one he heard.

Also, I don't recall if anyone has commented on Dana's remark that his "songs for capstan and falls" have "a chorus at the end of each line." It would be unrealistic to assume that he meant that was true in all cases, but I think we can assume that he meant it was most usually the case. That suggests to me a rather primitive shanty style, with one improvised line and a short repeating chorus (like "Haul Away, Joe!" or "Haul on the Bowline!"), rather than the more elaborate four-line rhyming stanza that we think of as the "classic" shanty form. "Round the Corner, Sally!" fits the earlier form. If that's what Dana meant, and the more elaborate form was still rare, it's a further suggestion that shantying was still in a formative stage in 1835.