Here's my summary of:
Garst, John F., "Poor Wayfaring Stranger--Early Publications." The Hymn, v. 31, no.2 (1980): p.97-101.

As popular as this hymn became in the late 19th & early 20th century, Garst found few versions in checking hundreds or 19th century hymn books.

The 1882 version above was published by Marshall W. Taylor & is the earliest he found that included a tune. He reported that he found another similar form of the text only from 1867*, but did not reproduce the words. The words varied enough that he thought it unlikely that the 1867 & 1882 versions came from the same source. Garst notes that while the 3rd verse with the 'children' chorus did not get into the hymnal versions, it was collected from oral tradition in the 20th century.**

Beaver's 'Going Over Jordan' from 1858 is clearly related but much simpler than PWS. Without other sources it's impossible to tell if the Beaver version is the "eroded & recomposed residue of a more formal text" or "a recording of an amorphous camp-meeting song that was recomposed to give the Taylor text."

Garst found nothing to confirm or refute the Odum attribution to Bishop Allen that Dicho mentioned.

The version of the tune that appears in most hymnals is the W.T. Dale arrangement in Cayce (see T's words). That version was arranged from Charlie D. Tillman's The Revival (1891) who transcribed it from his father's singing. Tillman's version is 4/4 and major; Dale's is 3/4 and minor with sharp 7ths. Taylor's 1882 version is major and has rhythms that are 'nonsense' but the basic tune fits with the more common version. The nonsense of the rhythms is 4/4 and note values that are very different from the usual.

There's more discussion in this thread.

*Revival and camp meeting minstrel. Philadelphia: Perkinpine & Higgins, 1867. p.272.

**Buchanan, Annabel Morris, Folk hymns of America. New York: J. Fischer [c1938] p. xxvii & 66.