(1) This is the link to the sheet music.

Title: A Wet Sheet & A Flowing Sea. A Nautical Song.
Composer, Lyricist, Arranger: Written by Allan cunningham. Adapted & Arranged by Thomas Walton.
Thomas Walton Publication: Philadelphia: John F. Nunns, 70 So. Third Street, 1837.
Form of Composition: strophic with chorus
Instrumentation: piano and voice
First Line: A wet sheet and a flowing sea, and a wind that follows fast
First Line of Chorus: Give me a wet sheet a flowing sea and a wind that follows fast
Performer: Sung with distinguished applause by Mr. Walton.
Engraver, Lithographer, Artist: E. Gillingham
Subject: Sailors
Subject: Bodies of water
Subject: Happiness
Subject: Storms
Call No.: Box: 122 Item: 107

(2) Two other sheets with a different melody at Levy are:

Title: A Wet Sheet and a Flowing Sea. [only one page is given]
Composer, Lyricist, Arranger: Written by Allan Cunningham.
Publication: Philadelphia: G.E. Blake, 13 South Fifth Street, n.d..

Title: A Wet Sheet and a Flowing Sea. [cover title A Health to the Outward Bound].
Composer, Lyricist, Arranger: [Words by] Written by Allan Cunningham. The Music Composed by M.S.
M. S. Publication: Baltimore: G. Willig, Jr., n.d..

(3) Two song sheets (without music) are at America Singing: Nineteenth-Century Song Sheets (American Memory, Library of Congress):

A wet sheet and a flowing sea! H. De Marsan, Publisher, 60 Chatham Street, N. Y. [n. d.]

A wet sheet and a flowing sea. Thos. G. Doyle, Bookseller, &c., No. 297 Gay St. near Ashland Square, Balt. [n. d.]

(4) MIDI & lyrics are also at Lesley Nelson's site (CLICK HERE).

(5) The poem is contained in Francis T. Palgrave, ed., The Golden Treasury (1875).

(6) There's a score with more simplified arrangement HERE (from Public Domain Music site).

(7) This song is sung in Tom Brown's Schooldays (1857) by Thomas Hughes (Part 1 - Chapter 6 - After The Match):

The sixth-form boys had not yet appeared; so, to fill up the gap, an interesting and time-honoured ceremony was gone through. Each new boy was placed on the table in turn, and made to sing a solo, under the penalty of drinking a large mug of salt and water if he resisted or broke down. However, the new boys all sing like nightingales to-night, and the salt water is not in requisition - Tom, as his part, performing the old west-country song of "The Leather Bottel" with considerable applause. And at the half-hour down come the sixth and fifth form boys, and take their places at the tables, which are filled up by the next biggest boys, the rest, for whom there is no room at the table, standing round outside.

The glasses and mugs are filled, and then the fugleman strikes up the old sea-song,

"A wet sheet and a flowing sea,
And a wind that follows fast," etc.,

which is the invariable first song in the School-house; and all the seventy voices join in, not mindful of harmony, but bent on noise, which they attain decidedly, but the general effect isn't bad. And then follow "The British Grenadiers," "Billy Taylor," "The Siege of Seringapatam," "Three Jolly Postboys," and other vociferous songs in rapid succession, including "The Chesapeake and Shannon," a song lately introduced in honour of old Brooke; and when they come to the words,

"Brave Broke he waved his sword, crying, Now, my lads, aboard, And we'll stop their playing Yankee-doodle-dandy oh!"

you expect the roof to come down. The sixth and fifth know that "brave Broke" of the Shannon was no sort of relation to our old Brooke. The fourth form are uncertain in their belief, but for the most part hold that old Brooke was a midshipman then on board his uncle's ship. And the lower school never doubt for a moment that it was our old Brooke who led the boarders, in what capacity they care not a straw. During the pauses the bottled- beer corks fly rapidly, and the talk is fast and merry, and the big boys - at least all of them who have a fellow-feeling for dry throats - hand their mugs over their shoulders to be emptied by the small ones who stand round behind.

~Masato