He should absolutely do whatever she says because he's awful at the game.* She managed to play the "I'm not a threat" card and parlayed her way out of the house's cross hairs. That's a tougher play for Cody because he's legitimately awesome at competitions.
But he has what amounts to 3 weeks of safety because production gave his girlfriend some power. There's no way she won that vote. She'd been given basically no positive coverage on the show. I could buy that the voters just wanted to throw a wrench into the season, but that's not really how these votes traditionally play out. It's not how they played out in the prior votes this season.
*He's not really that awful, he's just a #####. Getting Paul out was sensible, but not when you've aligned with him. It was a move that would have worked out better much later in the game, but it was too fast. That's a mistake that gets made every season, though. And it would have been fine if Cody knew how to talk to people or the producers hadn't given Paul immunity.