It may sound excessively obvious, but Shakespeare's a goldmine of songs that started out as folk. Examples:
Tomorrow Is Saint Valentine's Day
All a Green Willow Shall Be My Garland
I Was a Lover and His Lass
Western Wind
How Should I Your True Love Know
Jog On the Footpath Way
John Dory

also
Cherry Ripe (not the later art song)
Jolly Good Ale and Old
Roister Doister
A, Robyn
Martin Said To His Man (the right era, I think)
Greensleeves
The Three Ravens
King Henry the Fifth's Conquest of France
Fara Diddle Dyno
and, of course, To Celia (=Drink to Me Only With Thine Eyes)

Most of these have no surviving tunes, but tunes have been put to them over the yeas.   The Shakespearean ones usually appear in fragments, but the whole song is often known. Your local library, or Web libraries, should have a fair section under Renaissance Songs or some such search term.

You may also want to check current CDs under Elizabethan or Shakespearean songs...there are lots more.   Good hunting! Bob