Thank you, Lighter.

The Kentucky Harmony has been posted online by the University of Kentucky. A link to it can be found at the Wikipedia article "Kentucky Harmony." At page 32 is the tune "Judment" (sic) for a hymn whose first line is "My God what inward grief I feel." My sight-reading is rudimentary, but none of the parts for this four-part harmony tune seem to match the tune currently used for "I am a Poor Wayfaring Stranger." The tune would fit the words, but that is true of numerous tunes and hymns, and does not establish that they were ever sung together.

So my original question remains: When was the tune we currently use first published to the words of "I am a Poor Wayfaring Stranger"?

Addressing your BTW question, I'm not aware of other tunes being used for "I am a Poor Wayfaring Stranger," but nevertheless avoid calling any tune "the" tune used for a hymn, given that hymns were often sung to whatever tune fit (a necessity at a time when most hymnals lacked music).