Ravenscroft's books given in facsimile on the SCA Minstrel website contain the music for all the songs in them.

A text of "Lavendar's Blue" of 1674-9 is in the Scarce Songs 1 file on my website (Mudcat's Links). A claim that there was an earlier version was also made in an old thread, but so far no evidence for that has been produced.

There are very few songs in 'Pills to Purge Melancholy' that are earlier than 1621. [Tom o' Bedlam, and a few from Tho. Heywood's "The Rape of Lucrece", 1608. What are some others?]

All the songs (but without the music) in English songbooks with music), from Wm. Barley's of 1596 to John Attey's of 1622, with the exception of Thomas Campion's songs, and the madrigals, are reprinted in Edward Doughtie's 'Lyrics from English Airs', 1970.

Search the broadside ballad index on my website by year, 1580- 1620, where Stationers' Register entry dates are cited for all known broadside ballads. [Lord of Lorn, Ladies daughter of Paris (traditional), In sad and ashy weeds, Callino, In Summer time, In Peascod time.]

There are also Marlow's "Come live with me and be my love" and Raleigh's reply, and "How can the tree", and "How now Shepherd, what means that". See an anthology like Ault's 'Elizabethan Lyrics' for some others.

The music, where such is known, is located, but not given in Peter J. Seng's 'The Vocal Songs in the Plays of Shakespeare', 1967.

Songs without music printed with them are in:
A Handeful of Pleasant Delites, 1584
The Works of Thomas Deloney, by F. O. Mann [e.g., "The Fair Flower of Northumberland"]
England's Helicon [mostly poems].
Crown Garland of Golden Roses, by Richard Johnson, 1612.
Golden Garland of Princely Delights

There are some other books with songs mixed with poems, but one usually has to dig them out of bibliographies of 'Poetical Miscellanies', and there were songs in many plays, (chronologically arranged in Shownbaum's 2nd revised edition of Harbage's 'Annals of English Drama', 1964.] but I know of no bibliography of the songs, and most often no tune is known for them.