Joe_F, a nice version.

"Forsworn" here means- I'm not tied down; if he scarpers, I won't mourn.
So much of the poem is about not caring if "he" goes, that is is obvious that the gal does care very much.

Forswear, "to swear, to vow to bring about." An old sense of the word, also in the OED. Still used locally in the States, and I am sure in UK as well.

"Never hang your affections on a green-growing tree"- this idea of hanging the heart, affections, etc. on a tree appears in several songs. Must be a very old saying.