The OP contains two different songs, two different lyrics.
It is the second which attracts me to this thread.
Malcolm Douglas, of happy memory,
name-checks Bishop Thomas Percy and his Reliques of Ancient English Poetry, published in 1765
(London: 'printed for J. Dodsley in Pall-Mall').
Percy, the editor of that anthology,
references a folio manuscript in his possession for his source for "Balowe."

And hereby hangs a tale.
What Percy's publication does not disclose,
is that Percy made numerous alterations
to the text of this, as well as other, 'ballads and romances'
before printing them in his anthology.

So this post, and the next from this Mudcat member at this thread,
will offer two versions of the "Balowe" text to readers:
the first,
the well-known version from the 'Reliques' anthology,
representing Bishop Percy's changes and alterations;
and the second, from the scholarly work
Bishop Percy's Folio Manuscript. Ballads and Romances,
published in 1867 - 1868(London: N. Trübner & Co.),
with annotation and research by such scholars as
John W. Hales and Frederick J. Furnivall of Cambridge University,
and Francis James Child of Harvard University.

Lady Bothwell's Lament. A Scottish Song.
Percy's remarks say:
"In the Editor's folio MS., whence this song is printed,
it is simply intitled BALOWE."

Balow, my babe, ly stil and sleipe!
It grieves me sair to see thee weipe:
If thoult be silent, Ile be glad,
Thy maining maks my heart ful sad.
Balow, my boy, thy mither's joy,
Thy father breides me great annoy.

CHORUS:
Balow, my babe, ly stil and sleipe,
It grieves me sair to see thee weipe.   

When he began to court my luve,
And with his sugred words to muve,
His faynings false, and flattering cheire
To me that time did not appeire:
But now I see, most cruell hee
Cares neither for my babe nor mee. CHORUS

Ly stil, my darling, sleipe a while,
And when thou wakest, sweetly smile:
But smile not, as thy father did,
To cozen maids: nay God forbid!
But yett I feire, thou wilt gae niere
Thy fatheris hart and face to beire.   CHORUS

I cannae chuse, but ever wil
Be luving to thy father stil:
Whair-eir he gaes, whair-eir he ryde,
My luve with him maun stil abyde:
In weil or wae, whair-eir he gae,
Mine hart can neire depart him frae. CHORUS

Bot doe not, doe not, prettie mine,
To faynings fals thine hart incline;
Be loyal to thy luver trew,
And nevir change hir for a new;
If gude or faire, of hir hae care,
For womens banning's wonderous sair. CHORUS

Bairne, sin thy cruel father is gane,
Thy winsome smiles maun eise my paine;
My babe and I'll together live,
He'll comfort me whan cares doe greive:
My babe and I right saft will ly,
And quite forgeit man's cruelty. CHORUS

Fareweil, fareweil, thou falsest youth,
That evir kist a womans mouth!
I wish all maides be warned by mee
Nevir to trust mans curtesy:
For if we doe but chance to bow,
They'le use us than they care nae how. CHORUS

-- from Reliques of Ancient English Poetry, Volume the Second, Book II. Appears in Ancient Songs and Ballads, pages 194 - 196.