Traditional Ballad Index says the poem is by James Sloan Gibbons, NY Evening Post, 1862. It goes on to say at least eight musical arrangements came out, the most popular by L. O. Emerson. The American Memory website illustrates the sheet music or broadsides for a number of versions including those of other contenders, Warden, Morris and Bryant among them, and African-American versions as well.
Take your pick of authors for the music you prefer, but Gibbons has the best claim for the original words.