Infallibility of the prophets in Islam???

This post is more of a question than commentary. In various khutbahs (the Islamic version of a sermon delivered at Friday prayer), I've heard some comments about the various prophets that almost make me think that the prophets are infallible and don't make mistakes or sin. Although they are indeed guided and inspired by God, their humanity still dictates their imperfection. The Quran mentions Adam's fall from grace in the Garden of Eden and also mentions Moses' (Moosa, in Arabic) murder of an Egyptian slave driver, although those are the only mentions I recall of prophets' mistakes. These are proof to me that Islam doesn't propose their infallibility. However, as I've said, I've heard comments that make me think otherwise so I figured I'd pose my question here.

Christianity definitely accepts that the prophets were imperfect men and sinned. In the Bible, there are stories of Adam's fall, Noah's drunkenness, David's adultery, Moses' murdering, and I'm sure more. The prophets were guided and inspired by God but at the same time, they were flawed human beings. The only man claimed to be free from sin was Jesus because of his being simultaneously God and man. His human side was tempted but he resisted and remained sin free.

However, I've heard Muslims get offended at the prospect of David's adultery or Noah's drunkenness. Does Islam preach the perfection of prophets? I don't see how that could be considering the Quran's inclusion of Adam's fall and Moses' murdering. Does Islam claim that Muhammad was sinless? I don't think accepting his sin would have any affect on the religion. He could easily be divinely guided when delivering the Quran without error yet still be an imperfect, flawed man. Just because he as a man is flawed doesn't mean the Quran would have to be flawed.

As I said, I don't know what Islam's stance is on this subject. If anyone knows, please inform me. If not, I guess I'll just have to do more research on my own.