Parallel Anthology Track 16
This song is written from the point of view of Charles Guiteau, assassin of President James A Garfield, executed for this crime in 1882.
According to the Old Weird America blog, this song "originated as a printed broadside and have been taken in folk tradition ever since. It was written in a form of a 'goodnight', a criminal’s confession before his execution. [...] Guiteau’s 'goodnight' is based on a New York broadside 'The lamentation of James Rodgers', from which it borrows entire verses, just remplacing the name of the criminal and the dates."
Bob Waltz agrees, saying that "the unknown author simply took the earlier ballad 'James A. Rogers' (executed in New York in 1858) and 'zipped in' the details of the Garfield case. This is no surprise; the same tune carried at least two other murder/confession ballads, 'John T. Williams' and 'Ewing Brooks'."
The Roud Folksong Index classifies this song as a version of Roud 444. A Roud number search for this song  returns several collected versions which pre-date Harrell's recording, dating as far back as 1907 and from a wide geographical area (Missouri, N Carolina, Oklahoma, W Virginia).
The wikipedia page for this song describes it as "traditional" and "based on another old ballad, 'James A. Rogers'".
Given that the tune and most of the words appear to date back to at least shortly after 1858, with the remainder of the words being inserted shortly after 1882 by an "unknown author", and in the absence of evidence of any competing claim to authorship, there are good grounds for treating this composition as public domain.
Charles Guiteau by Ramblin' Jack Elliott recorded between 1952 and 1955 (according to album title): Spotify Unable to find whether Elliott claimed arrangement credit
Charles Guiteau by Virgil Sturgill recorded in 1955 (according to RFSI): Spotify Unable to find whether any arrangement credit is claimed for this version
Charles Guiteau by Mr. L.O. Smith recorded in 1960: Max Hunter Collection