For me, if the Giants disappeared I'd probably throw my lot in with the Packers. Old school, lots of history and tradition, lots of winning football, well run, conservative, no mascots (although they do have cheerleaders and we don't), and we gave them Lombardi. I find a lot of the attributes I like about the Giants to also be present in the Packers. I feel as if they are a kindred franchise, the closest thing to us in the league.
As for the head-to-heads, we've played the Packers in the NFL Championship game 5x and we've played the Bears 7x. Which helps explain why I don't feel warm and fuzzy about the Bears as I do about the Packers.