The "Mark Teixeira will only go where he has a chance to win!" stories have gotten a bit silly. Yes, winning is probably more fun than losing. That being said, $20M+ per year buys a lot of off-the-diamond fun, especially if you're the biggest star on a losing team and can add endorsement money that's not likely to go to the Levale Speigners of the world instead.
If he goes to the Orioles, he'd be joining a nice young core that includes Adam Jones, Nick Markakis, catching prospect Matt Wieters and more, leaving open the possibility that the O's could win sooner rather than later. Throw in the fact that they have a lot of interesting vets that are on the market and which could bring back some useful arms, and the O's could climb past the disarmed Blue Jayes and into contention in a hurry. In that situation, Teixeira also gets credit for having put the Orioles over the top, possibly cementing a Hall of Fame candidacy while he's at it.
If he goes to the Nats, he's the face of the franchise, joining Ryan Zimmerman and a talented young outfield. Odds of winning here aren't quite as good, but it's not like the Phillies have been making great moves this off-season, the Marlins have been shedding salary, the Braves are still short pitching, and the Mets remain Heimlichian until proven otherwise.
And while folks are fond of pointing out that A-Rod's signing with a losing team didn't work out so well, Pudge Rodriguez' did. It started a renaissance in Detroit that led the team to respectability, encouraged other free agents to sign there, and stood as a symbol for the franchise.