More precisely, the line runs "Although thy woes at first seemed double ...".

It takes some stretching to count the 1670s as Renaissance, but I guess they are not that critical on those Faires.

The original tune "The Willow Green" was by a professional composer for a theatre play; it seems to be lost. The lyrics (are these the original ones?) suggest a sad melody, with many Baroque affections. Obviously the tune became popular and thus subject to parodies like "My Love sleeps on another mans Pillow". "The willow green turned into white" is therefore sold as a "sequel" - the maiden's reply - to the same tune. A joyful tune would be more adequate, and we are free to invent one.

The poem "The Willow Green" specifies:

Come all that bears good will unto me,
do so much as tell me how,
This green garland doth become me,
which I am forst to wear now,
Because obdurate she doth prove,
whose beauty might become a Queen,
And most unfaithful is in Love,
which makes me wear the willow green.

In the illustration I fail to see such a garland - it might not be meant physical at all. Willow + green = sorrow with some hope left; this is a very common symbolism. English poets, in all degrees of sophistication, are particularly fond of the willow; one of the reasons being that it neatly rhymes with mill-o and many other -o's.

Desdemona's

The poor soul sat sighing by a sycamore tree,
Sing all a green willow:
Her hand on her bosom, her head on her knee,
Sing willow, willow, willow.

is as English as can be; the Italian translation "Salce, salce, salce" in Giuseppe Verdi's "Otello" just does not work. (The rest of this ingenious opera is quite true to Shakespeare and truly taking the story home to Italy, a must-hear for all music lovers including despisers of typical Italian operas.)