Fascinating, Matthew. By chance, this website

http://www.outoftheordinarymusic.com/ott.html

offers the lyrics of a "Pretty Peggy of Darby O" whose first line is identical to that in the Bodleian (but unfortunately not in the online Bodleian Ballads collection). It comes from a Boston, Mass., broadside:

There was a Gay Captain in Darby Town,
And he was come a Recruiting O:
A Pretty Chambermaid his Heart soon betrayed,
And she was called sweet Peggy O.
      And she was called sweet Peggy O.

O come down the stairs with your golden Hair,
O come down to me said the Captain O;
For I to Killarney must now haste away
Then bid me farewell pretty Peggy O.
      Then bid me farewell pretty Peggy O.

The Damsel replied you need teize me no more;
You are not the man to my fancy O;
I will not be ty'd to be a soldier's bride,
So you may be gone now for Peggy O.
      So you may be gone now for Peggy O.

Never heed her disdain said his Brother so bold,
She is not worth your regarding O;
There'll be fair girls enow who their hearts will bestow,
When you come to the Town of Killarney O.
      When you come to the Town of Killarney O.

Then over the ocean they soon sail'd away,
And safe they arrived at Killarney O;
Yet tho' so far away he was often heard to say
Here's a Health to the Pretty girl of Darby O.
      Here's a Health to the Pretty girl of Darby O.

Sweet Peggy still ran in the Captains mind,
No other he thought was so charming O;
Love alone fill'd his breast that he never could rest,
And he died for the Pretty girl of Darby O.
      And he died for the Pretty girl of Darby O.

I almost forgot: the original song sheet ("c1782")evidently includes the tune! And there's a nice sound clip too!

All in all this text and tune must be at scarcely more than one remove from the original.