Right. The song is a satire, and the ill-natured "she'd no more be a maid" is part of the attack.

On then: I woke up this morning, and the first thought in my head was the brilliant insight that the 'fains' are mixed up.

fain - from the Anglo Saxon, means 'gladly'.
feign - from the Old French, means to pretend, imagine, invent

Change my (and Touchstone's) first verse to:


I'm a heartbroken raftsman. From Greenville I came.
All my virtue's departed for a lass who did feign.
==============
Every time I encounter the word 'fain' I remember a deathless scientific poem I learned in grade school:

Scintillate, scintillate globule viscific.
FAIN would I fathom thy nature specific.
Distantly poised in the ether capacious,
closely resembling a gem carbonaceous.