I have vols 1-3 of McCurdy's 'When Dalliance was in Flower' series and 'Son of Daliance'(primarily from 'Pills to Purge Melancholy' and some from earlier drolleries and a few are from early 18th century single sheet songs with music). Two more records by him of such material are "Lyrica Erotica", vols. 1 and 3, on Prestige-International (13044 and 13050) [Vol 2 was Arthur Argo's bawdy Scots songs, 13048]. I've never carefully checked, but think McCurdy shortened some songs, but more for time curtailment than for the purpose of expurgation.
McCurdy didn't always use the original tune, in fact there weren't any known originals for some of the drollery songs he did.

We will return to the question of Elizabethan songs later.

"Kitt hath lost her key" on the 1st 'dalliance' volume never had a tune until McCurdy composed one. The song is one of John Payne Collier's forgeries that can be put at about 1846-7. [As noted in the Collier file on my website I've seen the original MS copy of it.] Volume 2 is all from 'Pills to Purge Melancholy' but some were printed earlier. ("Merchant and the Fiddler's' wife is on a broadside ballad by J[ohn] P[hillips] and "My Thing is my own" was from a broadside called "Trap, or the young lass". The oldest there is undoubtabley "Uptails All". The tune appeared much earlier than any text, being in the Fitzwilliam Virginal Book, the date of which is uncertain but seems to fall in the range 1620-35, although many tunes in it are much older.
Vol. 3 is mostly from 'Pills to Purge Melancholy'. "Pillycock" is in Pills, iv 311 (1719) but came originally from Thomas Heywood's 'The Rape of Lucrece', 1608. "Celia" is from Pills which doesn't note that the song was by Tom Brown (sometime after 1680). "Old Brass to mend" is also from Pills but there are earlier broadside versions (but after 1655). "The Hive of Bees" is from Pills, but also appeared in earler drolleries. I've got a much earlier version from a manuscript that's only a few years later than 1603 in the Scarce Songs 1 file on my website. I think the rest in this volume are later than about 1670. Son of Dalliance: The oldest are "I dreamed My Love", c 1643 (Percy Folio MS [c 1643]: Loose and Humorous Songs) [This is also found in a couple other manuscripts, but I have no estimated date for them.] For the short history I have of "She lay all naked", apeared in 'Wit and Drollery, 1656. I've also seen it in a few manuscript, but not assuredly older than 1656. See Scarce Songs 1 on my website, for a broadside expansion [Almost forgot; it's now available on the Bodley Ballads website]. "A Maiden's Delight" is from 'Merry Drollery'. I've forgotten exactly where "Three Birds" came from. It's in a rare drollery of the 1660's or early 1670's. The rest are later, the last, "The Fornicator" being by Robert Burns. Lyrica Erotica 1. The oldest seems to be "Walking in a meadow green" from the Percy Folio MS: Loose and Humorous Songs. I think all the rest are later than 1660. Lyrica Erotica 3: The oldest is "A Puritan of Late" which was probably from a drollery or the Giles Earle MS, c 1625. The odd 4 1/2 line stanza is unique for songs to "Last Christmans 'twas my Chance", which is in Pills to the wrong tune. The original tune is of 1622 or 1623 and an ABC of it is among the broadside ballad tunes on my website.

It looks like there might be about 3 songs among the above that might be a little earlier than the earliest record we have of them, and be Elizabethan, but we have none that can be proven to be such on the basis of presently available information.

Some other recordings:
There's a selection of songs from 'The Merry Muses of Caledonia' by Paul Clayton [Worthington], Elektra 155. Another record by Clayton of bawdy folk songs is 'Unholy Matrimony', Elektra 147.

Ewan McColl also did a record of selections from 'The Merry Muses of Caledonia', for Folk-Lyric if I remember correctly, but it wasn't sold over the counter and I never found a copy.

One side of Richard Dyer-Bennet's 'Mark Twain's 1601) was bawdy folk songs (Richard Dyer-Bennet Records, 1962, no id number)

I was stupid enough to get one of Oscar Brand's bawdy[?] ballads records, but smart enough not to repeat that mistake.