Bill, neither of the scores you linked presents the air that Eberhart and the Boarding Party used for "The Shanghaied Dredger", nor is it "The Irish Exile" printed in O'Neill's Music of Ireland (O'Neill, Francis & O'Neill, James [1903] "O'Neill's Music of Ireland". Chicago: Lyon & Healy.), as is sometimes suggested.

The association of "True Irish Hearts" with "The Exile's Lament" is noteworthy because it might allow us to trace the song's introduction to Baltimore. I was able to find a listing for "True Irish Hearts" in the Baltimore Sun as early as 1887, and the piper James T. Touhey fell down an elevator shaft during a Baltimore run of the show in 1890.

The broadside "The Shanghai Dredger" was published by Baltimore printer William J. Schmidt, presumably between the 1886 publication of "The Exile's Lament" and Schmidt's death in 1911. There is a broadside titled "The Irish Exile" printed by A. W. Auner of Philadelphia here which is "The Exiles Lament" composed by J. F. Mitchell in all but title. Whether this predates the "The Shanghai Dredger" printed by Schmidt is yet unknown to me, but it's possible that one printer copied the other.

I'm still trying to discover whether the Mitchell air was retained in the oral tradition in Maryland or Virginia. I'm hoping to have an answer before long.