What you are looking for is "Christmas song partner songs." Partner songs are what you call two songs that have exactly the same chord changes, so that you can have two people (or two groups of people) sing them (or play them) at the same time, resulting in some nice harmonies and countermelody effects.
Choirs sing "partner songs" all the time (something I learned in previous jobs I've had where I taught both band AND choir. If you do an internet search for something like "Christmas partner songs," you will find some results for what you are looking for. Here's a good list of these that I found with a quick search: http://musicedresources.weebly.com/35-partner-songs--rounds.html Not all of the partner songs listed there are Christmas partners, but a good number of them are. If you search some more on the internet, you can probably find even more lists of partner songs like this.
"Rudolph The Red-Nosed Reindeer" and "Have a Holly, Jolly Christmas" have exactly the same chord progression, and therefore this is a case of 2 partner songs that are BOTH Christmas songs. (Both songs were written by Johnny Marks.)
The way that choirs usually perform partner songs (and what would also probably work great for your jazz combo) is to first have everyone sing (play) song #1 all the way through, then immediately everyone then sings (plays) song #2 all the way through, then divide up the ensemble so that half (or some) are singing (or playing) song #1 while the other half of the group are singing (or playing) song #2.