If the music for the song is out of copyright (or in the public domain), then you can indeed write and copywrite your own arrangement, lyrics, and performance of the song, and sell it without fear.
If the music is not out of copyright, then you can…
If the song is not Public Domain, then you cannot record it, and certainly not sell it, without permission, even if you change the lyrics and perform it yourself. But many Christmas songs are public domain, so check to see.
Last christmas a friend of mine wrote alternative lyrics to a well known classic christmas song. I plan on compiling about 10 or 15 songs altogether, including the one she wrote, and putting them onto a CD for her gift this year. I can't sing so I p…