"ONE MORNING IN MAY" has been collected numerous times in North America. Sandburg, 1927, The American Songbag, gives two versions from the Appalachians, pp. 136-138.
Vance Randolph, in Ozark Folksongs, 1980, vol. 1, pp. 266-269, gives five versions. A short variant collected in Missouri in 1930 emphasizes the fiddle:

C
They had not been there scarce an hour or two,
When out of his satchel a fiddle he drew,
An' he played her a tune caused the valley to ring,
Hark, hark, says the lady, hear the nightingale sing.

Oh now, says the soldier, it's time to go home,
Oh no, says the lady, just play me one more tune,
For I'd rather hear the fiddle, one touch of the string,
Than to see the water glide, hear the nightingale sing.

He tuned up his fiddle to one higher key,
An' he played her the same tune right over again,
A-causin' the valley to echo and ring,
Hark, hark says the lady, hear the nightingale sing.

Oh soldier, oh soldier, won't you marry me?
Oh no, says the soldier, that can never be,
I've a wife in Furlando with children twice three,
An' two in the army's too many for me.

Another version, also collected in Missouri, concludes with the usual warning to fair damsels that ends so many songs:

E
Come all ye damsels, take warning from me,
Don't place your affections on a soldier so free,
He'll love you and leave you and give you no ring,
For to rock your own cradle while the nightingale sings.

Come all you young damsels, take a warning from me,
And never lay down 'neath a green willow tree,
My cheeks were once red as the bud of a rose,
But now they are white as the lily that blows.

It was commonly played as a fiddle tune.