Do AL teams ever let a pitcher hit and use the DH for a position player?

It does sometimes happen. Let's say the Yankees have Jorge Posada DHing and Cervelli gets hurt and they decide to run for Russell Martin in extra innings leaving them with no catcher, so they have to use Posada as the catcher. The DH is now lost and all the pitchers either have to bat for themselves or there has to be a new pinch hitter every time that the pitcher comes up, just like in the National League.