Tune- "Well-a-Day(e)!" Can be found in Chappell's "Popular Music of the Olden Time," 1859, vol. 1, p. 176, and "Essex's Last Good Night," pp. 174-176.
Can' find the Carol in Mudcat.
The Carol uses the tune, but otherwise has nothing to do with "Essex's Last Farewell."

Lyr. Add: Christmas Hath Made An End
(Source- Bullen, A Christmas Garland)

Christmas hath made an end,
Well-a-day!! Well-a-day!
Which was my dearest friend,
More is the pity!
For with an heavy heart
Must I from thee depart,
To follow the plow and cart
All the year after.

Lent is fast coming on,
Well-a-day! Well-a-day!
That loves not anyone,
More is the pity!
For I doubt both my cheeks
Will look thin from eating leeks;
Wise is he then that seeks
For a friend in a corner.

And our good cheer is gone,
Well-a-day! Well-a-day!
And turned to a bone,
More is the pity!
In my good master's house
I shall eat no more souse,
Then give me one carouse,
Gentle kind butler!

It grieves me to the heart,
Well-a-day! Well-a-day!
From my friend to depart,
More is the pity!
Christmas, I fear 'tis thee
That thus forsaketh me;
Yet till one hour I see,
Will I be merry.

www.hymnsandcarolsofchristmas.com
Source: A. H. Bullen, 1885, "A Christmas Garland," John C. Nimmo, London, pp. 244-245.