Green’s Dictionary of Slang

faint adj.

(US) a euph. for drunk.

1851
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a.1856
[US]Kalida Venture (OH) 11 Apr. 2/4: Drunk [...] faint.
[US]Burlington Sentinel in Hall (1856) 461: We give a list of a few of the various words and phrases which have been in use, at one time or another, to signify some stage of inebriation: [...] faint.