Let the record reflect that I do not scrunch my whole face, just my nose, but only when it's used as a slang word for urinate. Inconsistent, but there you go.
I actually have no trouble with "I was p*ssed off" = I was angry or "That really p*sses me off" = That really makes me angry. (I don't use it around my mother, but even at work, that's okay. But "I need to go take a p*ss" is not okay.)
Americans don't (in my experience) use this phrase to mean "to leave" with the exception of "Oh, p*ss off!" meaning "Oh, go away."
I have never heard an American say "He p*ssed off about an hour ago" or "If you say that to him, he'll p*ss off."