Katie Couric threw a Christmas party for 160 of her “CBS Evening News” staff — and footed the whole bill herself after singing karaoke. A source said of Couric’s annual bash at Jake’s Saloon in Chelsea the other night, “There was dancing, and everyone got into the karaoke. Katie, who sang a medley of songs, paid for the whole thing to thank them for all their hard work.” Meanwhile, her ABC rival, Diane Sawyer, hosted her own bash with Sam Champion at the Ainsworth in the Fashion District. A source told us, “They both paid for an open bar and food with a DJ for four hours.”