Of the earlier songs mentioned here, "The Ocean Burial" by George N. Allen, 1850, is in the DT.
"The Sailor Boy's Grave" from 1841 is in Levy Sheet music. The copy is hard to read.

THE SAILOR BOY'S GRAVE
Written and composed by J. Martin, Esq. of Clifton

Oh bury me not in the dark cold grave
With the rank weeds growing o'er me,
Let me sleep 'neath the silent wave,
The sea-nymphs watching o'er me.

I ask no proud marble to mark the spot
Where the sailor boy is sleeping,
He is not there alone, he will not be forgot
Where the mermaids are nightly weeping.
*He is not there alone, he will not be forgot
*Where the mermaids are nightly weeping.

No- the spirits that hover o'er the deep,
Even in the silent night,
Will pause by his grave, but not to weep.
When the stars are shining bright
There'll be nought to disturb the stillness there,
But the night wind gently driven
Or the low flutter of the sea-fowl near
Or the distant echo given.

The flowers may bloom and the gay birds sing
Where the Cypress waves its head,
But what care I if the violets spring-
Are they heeded by the dead!
Then bury me not in the dark cold grave
Where the worms their vigils keep.
Let my winding sheet be the ocean's wave-
Oh! 'tis there I'd wish to sleep.

* Repeated in the sheet music but not in the broadside. Clifton Pub., Baltimore, Geo. Willig Jr., 1841, sheet music. Sailor Boy's Grave
(http://levysheetmusic.mse.jhu.edu/cgi-bin/display.pl?record=042.117a.000&pages=3)

Also printed in broadside by T. G. Doyle, Baltimore (American Memory).