The modern sense of screwed originates in the mid-1600s with a sense of to screw as a means of "exerting pressure or coercion", probably in reference to instruments of torture (e.g. thumbscrews). It quickly gained a wider general sense of "in a bind; in unfortunate inescapable circumstances". When the verb screw gained a sexual connotation in the early 1700s, it joined the long-lasting association of sexual imagery as a metaphor for domination, leading to screwed gaining synonyms like fucked and shagged. On a more general note, this is a prime example of the frequent tendency for verb participles to evolve into adjectives.
The sense meaning "intoxicated" is from the early 1800s, and is associated with the term screwy, and the idiom to have a screw loose.
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