These songs containing references to St Patrick's Day are quoted in Mick Maloney, Far From the Shamrock Shore. Wilton, Cork: Collins Press, 2002
[by the way. the book, subtitled "The History of Irish-American Immigration through Song", comes complete with a CD. These two songs aren't on the cd, however.]
another book of interest would be Edward Harrigan and David Braham Collected Songs Edited by Jon W. Finson

ST PATRICK'S DAY PARADE
Ed Harrigan

Then here's sucess to Patrick's Day though it comes but once a year
And though I'm not a drinking man, on that day I'd drink my beer.
I'll wear the shamrock in my hat, the green my boys 'twill never fade
And march along with Dan and Pat in the Patrick's Day parade.
We shout hurrah for Erin Go Bragh and all the Yankee nation
Stars and stripes and shamrock bright arrayed
The Irish shout, the girls turn out to see the celebration
We march stiff as starch in the Patrick's Day parade.

IF IT WASN'T FOR THE IRISH AND THE JEWS
Jerome and Schwartz

Talk about a combination, hear my words and make a note
On St Patrick's Day Rosinsky pins a shamrock on his coat
There's a sympathetic feeling between the Blooms and MacAdoos
Why Tammany would surely fall
Ther'd really be no hall at all
If it wasn't for the Irish and the Jews.

(recorded by Billy Murray in 1912
that page of Victor recordings also mentions St Patrick's Day Jig recorded by Victor Military Band, 1911)