Correction related to post below this one.
Hmmm. It seems there are public domain Xmas carols, after all. Quite religious, most of them, making them useless to me, but there are a few hopefuls in the list. There's light at the end of the tunnel, after all. The following info comes from easysonglicensing.com. I'm greatly relieved that I can use a few songs in the book.
Angels from the Realms of Glory
Here We Come A-Caroling
Jolly Old St. Nicholas
Joy to the World
O Come All Ye Faithful