Compare the 1839 poem by Chapin with "The Ocean Burial" published in 1850 by Oliver Ditson (and in the DT). It is called a "new and improved edition."
The sheet music cites George N. Allen as composer of the music, but does not name the author of the lyrics. Sheet music at Levy.

In 1857, Ditson published sheet music in guitar arrangement by "Converse."
H. De Marsan published "The Ocean Burial" as a song sheet, but without attribution.
Both the above in American Memory.

The Traditional Ballad Index correctly cites Chapin 1839 as author (Southern Literary Messenger), but mistakenly says the "entire thing" is credited to Allen in the 1850 sheet music. Allen is credited with the music in the Ditson sheet music, but he is not cited as author of the poem; no author is cited.