I'm pretty certain it's an old song dating back to the late 17thc.

There's a verse in Gale Huntington's 'Songs the Whalemen Sang' p274 under the title 'Moll Brooks' It's possible 'Moll Brooks' is a Mondegreen for 'Marlborough' which is well-known tune. 'Moll Brooks' is the designated tune in the journal (Euphrasia, 1849)

I've lost my beau but I care not
He would come back but he dare not
I could have another but I will not
I will be happy and free.

The Greig Duncan Collection Vol6, p192 has a slightly longer version recorded in 1906.

I've lost my love an I keena well fu
I've lost my love an I carna
For laith would I be jist to lie down an dee
An to sit down an greet would be bairnly.

O plague on the men their so sly an demure
As pawky as diels wi their smiling
As fickle as winter in sunshine an shower
The hearts o' fair maidens beguiling

3rd starts as 1st....
......................
The lossing o ane's thew gainin o twa
Fain wid he come back but he daurna.

Cecilia Costello of Birmingham sings the one verse fragment on the double CD recently produced by Musical Traditions (MTCD363-4) of all her songs. It's on disc 2. track 60.

Oh, I lost me love and I care not,
He wants to come back but he dare not
So I'll have another as good as the other
And I don't care for neither.

Notes: 9 entries in Roud Index. John Strachan and Ray Fisher knew it in Scotland and the Opies found it as a children's game song in Birmingham.