I gather you mean then songs from Shakespeare's period, rather than Shakespeare's songs.
Songs from all the English songbooks, 1597-1623 are in E. Doughtie's 'Lyrics from English Airs', except for those of Thomas Campion and the madrigal books. He doesn't give the tunes, but most are fairly easily found. Several are among the ballad tunes on my website. Lute ones of the 16th century are in John Ward's 'Music for Elizabethan Lutes', 2 vols., 1992.