A song "Molly Malone" was published in NY by Pond & Co., 1870. This 'sweet Molly' never resorted to peddling mussels. Words by Thomas Cory, Music by J. G. Maeder.
"Sweet Little Molly Malone" came out in 1885. "Mollie Malone" appeared in 1875.
These are not the same song. Or Molly had several persona.