Comparing the lyrics, as given earlier, shows us that there are strong similarities, but it looks to me like the second and fourth lines don't scan the same. It'd be hard to sing "We Shall See Her..." to the tune of "Sunny South," in particular the fourth line. It looks to me like someone knew We Shall See Her and, perhaps casting about for lyrics to put to a nice tune, grafted them onto the tune, made the necessary rhythmic changes and ended up with what we call Sweet Sunny South.

WE SHALL SEE HER NO MORE:
(Oh, the bright sunny South, where the sugar cane grows,
And the cotton flow'rs gracefully bend in the wind
Where life, like a dream, full of happiness flows,
And Massa, dear massa, was always so kind)

THE SUNNY SOUTH:
(Take me home to the place where I first saw the light
To the sweet sunny south take me home
Where the mockingbirds sang me to rest ev'ry night
Oh, why was I tempted to roam?

Was it likely 19th C. plagiarism? I'd bet on it. If they were both completely folk songs, and not produced for the popular-music "trade," then the plagiarism might be more "unconcious," although the folk do indeed steal words (and tunes). Though, to be honest, to the folk, it's less like stealing and more like borrowing, since the idea of ownership of a song is, well, "different," shall we say? I don't want to say that folk performers don't think of songs as "owned," because they do, but ownership is more an attribute of performing than of creation. Traditional performers can "own" Barbara Allen, or any other ancient song, but can create a song or tune and let it go "out there" without qualms. This is because, in smaller communities, performers get known for their renditions, their versions, and it's not meant that they created it when it's called "their song."

But I think the creator of Sweet Sunny South knew that he/she hadn't created the lyric from whole cloth, and probably relied on the weaker copyright laws and practices that were in use back then.

Songbob