According to Wikipedia, weekly Password airings were added to the daily weekday shows starting on Dec 25, 1966, which was a Sunday. Allen Ludden starts the show by saying that this is their first Sunday show, so I think it's the Dec 25 show.
Not sure why people are saying Happy New Year rather than Happy Holidays, but maybe people weren't saying Happy Holidays in 1966? Perhaps the Sunday versions were supposed to start on Jan 1, so when they pre-taped this episode, maybe they thought it would air on Jan 1 rather than Dec 25. Although I don't think there was a football game on Jan 8. The AFL/NFL championships were done by Jan 1. But one was played on Jan 1, so that would be a "football game next week" for Dec 25.