The original title was "Melancholy." The sheet music at the Levy Collection is:

Title: Melancholy.
Composer, Lyricist, Arranger: Words by Geo. A. Norton. Music by Ernie Burnett.
Ernie Burnett Publication: Memphis: Theron C. Bennett Co., 1911.

The title was changed to "My Melancholy Baby" in 1914. Five recordings (Jimmy Noone's Apex Club Orchestra; Paul Whiteman Orchestra; Dorsey Brothers Orchestra; Ted Lewis and His Orchestra; and The Little Chasers) [Realaudio] can be heard HERE.

~Masato