MY fav really is a "Yuletide" carol.....it heralds the New Year.
It's maybe 15th or 16th C. (i.e., Renaissance) New Year's Day carol and the music may remind of you of Greensleeves,which is also 16th C. composition (the Christmas lyrics came much later).
I have heard it sung to Greensleeves but didn't care for it.
But whatever the origin of the music of this version I'm a sucker for any carol in a minor key.
The best performance (imo) is by Apollo's Fire, (soprano Sandra Simon).
The Old Yeare Now Away is FledLyrics: (I've only listed 4 of the 6 Stanzas --1,2,3 and 5--- because the recording only has those 4. I especially like the last one.....

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The Old Yeare Now Away Is Fled
1. The old year now away is fled,
The new year it is entered;
Then let us all our sins down tread,
And joyfully all appear.
Let's merry be this holy day,
And let us now go sport and play,
Hang sorrow, let's cast care away
God send you a happy year.
2. For Christ's circumcision this day we keep,
Who for our sins did often weep;
His hands and feet were wounded deep,
And his blessed side, with a spear.
His head they crowned then with thorn,
And at him they did laugh and scorn,
Who for to save our souls was born;
God send us a merry New Year!
3. And now with New-Year's gifts each friend
Unto each other they do send;
God grant we may all our lives amend,
And that the truth may appear.
Now like the snake cast off your skin
Of evil thoughts and wicked sin,
And to amend this new year begin:
God send us a merry new year!
5. Come lads and lasses every one,
Jack, Tom, Dick, Bessie, Mary and John,
Let's carve the meat unto the bone,
For welcome you need not fear.
And here for good liquor we shall not lack,
It will whet my brains and strengthen my back;
This jolly good cheer it must go to wrack:
God send us a merry new year!