A British connection may exist through the ballad "Fond Affection," or "Once I Loved With Fond Affection." Brown, North Carolina Folklore, has several versions. Vol. 4 has the music and vol. 2 the lyrics. Version E collected in 1919 begins:
Once I loved a fond affection,
And he thought the world of me,
Till some dark-eyed girl persuaded;
Then he thought no more of me.

Go and leave me if you wish to,
Never let me cross your mind.
If you think I'm so unworthy
Go and leave me, never mind.

Coll. in Randolph Co., 1919, from Austin E. Elliot.

Version N starts:
I will pawn this watch and chain, love,
I will pawn this diamond ring.
I will pawn this heart from my bosom,
Hoping it brings you back again.

Go and leave me if you wish to,
Never let me cross your mind.
If you think I'm too unworthy,
Go, my darling, I don't mind.

Sung by Bascom Lamar Lunsford, Buncombe Co., no date.
Without vol. 2, I cannot give full lyrics. Five somewhat variant tunes are represented.

Other songs, considered as separate but with similar themes and (to me) that seem to be part of the same cluster, are "You Are False, But I'll Forgive You," "We Have Met And We Have Parted," "The Broken Engagement," "Broken Ties," "Parting Words," and several more.
The variations seem infinite.