I'm dubious of a relation to "Marching Through Georgia". Granted, MTG was hugely popular, and Thorp said Rooney wrote "Top Hand" in 1877, only 12 years after the Civil War ended. But other cowboy songs were derived from MTG, and they reflect this in their tune, meter and in having a chorus with a similar pattern of repeated phrases--I saw a couple just earlier today while thumbing through Lingenfelter's Songs of the American West. The only resemblance in the above fragment is the phrase "sing you a song" and the rhyming "along". Not even the meter is close enough to suggest a direct derivation, and the metrical patterns of both are ubiquitous.

Indeed, the catchiest, most distinctive part of MTG is its chorus; I don't recall a single song known to be derived from MTG which omits some variant of it. Even if the originator for some reason only used the verse tune, later cowhands would surely have supplied a chorus to complete it. The tune of MTG is not like "King of the Cannibal Islands", where the first part is most distinctive, and was often used by itself in the many derivations.

Also, a very common way to start a folk song in any genre is something like "[Come all you cow punchers] and I'll sing you a song, / [It's about my horse, Chopo] and it won't take me long..." Rhyming to "along" is also typical. In other words, the resemblance you detected is a formulaic one, and the lack of other similarities argues against derivation.

Still, it's a possibility, and why not derive a tune from MTG to use with these lyrics, if you feel so moved?