A PDF score for the tune, from the Kentucky Harmony supplement, can be found here under the title Baltimore (#53):
http://www.shapenote.net/berkley/SKyH3.htm [PDF]
The text is different ("Lord, I am vile! What shall I say?"— Isaac Watts). A PDF of the entire set of scans for the Supplement may be found at BostonSing.org:
http://www.bostonsing.org/music/suppl-kt-harmony/

Tim Eriksen YouTube clip (solo voice and fiddle):
http://www.youtube.com/watch?v=5JMCaKufxoY

The first marriage of the text and tune that I've found is in The Southern Harmony and Musical Companion (William Walker, 1835), "Garden Hymn", p. 90. Several full copies of this book are available online (for example, here), including a downloadable PDF at Google Books. Eriksen's clip notes mention some longer clip that also talks of his discovering an 1836 shape-note book, but I haven't seen the longer clip, so I don't know if that book is directly related to this song. If so, I suspect it was Walker's Southern Harmony.

Is the Nashville tune the same one we're discussing, (i.e., the tune sung by Smiley/Swearingen), or is it a different tune set to the same text? I can only view the lyrics. Fasola.org credits the words to J. Leavitte, Christian Lyre, 1830; but while this may be their cited source, I've found the text published as early as 1820 (The Providence selection of hymns: supplementary to Dr. Watts, p. 185, Hymn 228, 4 verses), and a nine-verse version, labelled "Pentecost Hymn" was one of two hymns printed at the end of Meditations Among the Tombs, by James Hervey in the "new edition" of 1824. The book itself had been published at least by 1794, though I don't know if the hymns were included in it originally. I also don't know whether Hervey wrote the hymn texts, but the fewness of them suggests he did (as well as others; the section heading is "Selected Hymns"). All the earliest hymn collections give no author or state "anonymous".

In addition to Hervey's nine verses, one version adds this final stanza, which is also found in some shorter versions:

10. There on that peaceful happy shore,
We'll sing and shout our suffering o'er,
   In sweet redeeming love.
We'll shout and praise our conq'ring king
Who died himself that he might bring
   Us rebels home to God.