Yes, you can say "I screwed up or I messed up." "I failed" is a more straightforward phrase. I wouldn't say that 'screwed up' is rude but it's slang; many people say it in my country and it's not considered rude. But you wouldn't write it in a formal exam! You could use it in everyday speech with no problem. It would sound as if you really know how to speak conversational English!

They all mean the same thing but screw up is a bit offensive or vulgar.

Technically, “screw” is a euphemism: “screw up” is a softer way of saying f-up. Same thing with the insult “screw you.” It’s quite informal. Messed up is the same meaning if a bit less intense, and without the hint of vulgarity.

Both are informal ways to say you made mistakes or did something wrong. While not really a bad phrase to say screwed up is a bit more crass a statement than messed up but since both would best be used in informal settings not much difference.

"Screw you" is definitely very rude and not advised to use! Different from "I screwed up" which is quite acceptable in informal conversation.