Why didn't they check David's immunity?

Laziness, incompetence....whatever you want to call it. Romero has always been keen on the idea of governmental/military futility, that they will chop down an entire forest in order to save it. I think its a rebelliousness against any organizational authority that is very much part of the 60s Counterculture to which Romero firmly belonged (a "movement" whose own attitudes lead to its own brand of stupidity, it should be noted).

Take from that concept what you will, but in the movie, there's no real specific motivation on the part of doctor to dismiss David. He's preoccupied with other matters, and knows that while an immune human may exist, the chances are so remote that he can afford to skip a few patients here and there. Now, why David doesn't speak up and tell them he's immune is beyond me.

I tend to think that it was because David had shot some of their guys, so they figured he must be crazy, and therefore was obviously not immune.

Look at David's expression in his face at the end. He looks crazy. He went crazy. But not because of Trixie, but because he lost wife and child.