Seems like there are two or maybe three "Old Paint" songs, though they are related in meter and (to a lesser extent) in melody.

"Goodbye, Old Paint" is the song John Lomax first printed in the 1916 (not the 1910) edition of "Cowboy Songs." It characteristically features the chorus, "Goodbye, Old Paint, I'm (a-)leaving Cheyenne."
The version performed by Texan Jess Morris (well presented in Tinsley's book, based on Morrisss recording for the Library of Congress)is presumably the earliest in substance, since Morris said he'd learned it from Charlie Willis around 1870. How much, if anything, Morris may have altered the "original" text over the years is, naturally, unknown.

Morris's version may vary just enough in text and tune from the 1916 Lomax version to be considered a "different song." But this is an entirely subjective judgment.

A clearly different song is generally called "I Ride an Old Paint."
Sandburg's "American Songbag" of 1927 provides the first known printing. As Q and others have noted, Sandburg says he got this song from Margaret Larkin, who got it from Lynn Riggs. Riggs was the ranch-bred Oklahoma author of the hit Broadway musical "Green Grow the Lilacs," which provided the acknowledged basis for Rodgers' and Hammersteins' '40s show (and '50s film) "Oklahoma!" Though not a professional composer, Riggs was undoubtedly capable of creating "I Ride an Old Paint" all by himself, based on his familiarity with Western themes; Sandburg's report that Riggs and Larkin had both learned it from an unknown "buckaroo...heading for the Border" is best taken with a grain of salt. It might be true or not, or partially true; Larkin and Riggs together may have fleshed out some cowboy's refrain.

We'll probably never know about that, but the significant point is that "I Ride an Old Paint" can't be firmly dated before 1926 or '27, and is first reported from Santa Fe, N.M.

I'll pick up this thread later this evening. Gotta go!