In a manner characterized by excessive drinking or drunkenness; in an intemperate way.
From crapulous with the adverbial suffix -ly, creating an adverb form. English regularly transforms adjectives describing moral states into adverbs to describe actions performed in those states.
You can describe someone as 'crapulous,' but when they're acting that way—stumbling, slurring—they're doing it 'crapulously.' This distinction between state and action is subtle but shows how English uses adverbs.
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