Both are valid english, while obviously colloquial.
The meaning is similar in this context, but not quite the same. Used as an adjective, "fucked up" means "in a mess", or "in a bad state", whereas "fucked" means "irreperably messed up", "severely broken" or "doomed".
In this sense, "fucked" is worse than "fucked up". "Fucked up" implies that it could still be fixed, whereas "fucked" implies that it's beyond repair.
The "doomed" meaning of "fucked" can be seen in various contexts - eg on learning that your company was probably going to go bankrupt, you might say to your colleague "Man, we're fucked.", meaning "We're doomed - we can't recover from this."
"Fucked up" (adjective) can also mean "severely intoxicated", from drugs or alcohol, and in fact "fucked" can be used in this context too, eg "I can't even remember getting to the club, I was so fucked by that point".
And, as you say in your first sentence, "fucked up" the verb can be subsituted for "messed up", meaning "made a mistake", in any context, so you could describe making a mistake like "I really fucked up last night, I'm sorry", meaning "I made a terrible mistake last night, I'm sorry."