Check out William Billings, Boston composer during the Revolution. Some say that he composed the first carol in American, a rendition of The Virgin Unspotted. Bethlehem is his fuging tune setting of While Shepherds Watched. There are also serveral other compositions with a Christmas theme. Billings was also composer of the Revolution's number one tune, more popular than Yankee Doodle: "Let tyrants shake their mighty sword." The title like everything else from that period is the name of a town. There are also many contemporary singing masters who produced some Christmas music. One thing to be considered here is that the celebration of Christmas was not big, coming out of the Puritanical heritage.
Also check out Ruth Crawford Seeger's American Folksongs for Christmas. I don't believe that any of them are assserted to be in existence during the Revolutionary War, but there is also little reason to believe that many of indigenous Appalachian mountain songs were not sung during the 1700's. The spirituals may also have been widely sung during the 1700's