It is a more acceptable way of saying "I'm f****d", which refers to the sexual act, literally, but which has devolved into the meaning that you wrote "messed up". If you screw someone in a deal or exchange it means you somehow cheated, gypped, conned or duped them.

we use it daily in the US. there are different meanings... the traditional meaning is to twist in a screw (noun) using a screwdriver. another way to use it is: to screw-to have sex, but dont use that term around your teacher as it is kind of racy. "im screwed" comes from the sex definition...means "there is nothing i can do to save my self in this situation" (this is not racy) to screw up-to mess up, to do something wrong