In the manner of someone who is drunk; stumbling, slurring words, or acting recklessly due to alcohol intoxication.
From 'drunken' (past participle of archaic 'drink' in Middle English) plus '-ly' adverbial suffix. The form 'drunken' originally meant drunk but became archaic for the state, leaving 'drunk' for the adjective and 'drunkenly' for the adverb.
It's oddly telling that English needed the adverb 'drunkenly' to describe behavior—suggesting drunken actions were common enough to warrant their own descriptor, which actually appears in texts going back to at least Shakespeare's time, showing alcohol's long prominence in English culture.
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