Why is a drunk person often said to be "cut" or "half-cut"?

Solution 1:

It appears to be a short for “cut in the back/leg” as suggested by Green’s Dictionary of Slang:

cut adj.1 - [abbr. cut in the back under cut v.2 ]

• drunk; thus half-cut adj.2

cut in the back (adj.) (also cut in the eye, ...leg) [fig. use of SE]

• very drunk.

• 1650 [UK] *Eighth Liberal Science n.p.: No man must call a Good-fellow Drunkard [...] But if at any time they spie that defect in another, they may without any forfeit or just exceptions taken, say, He is Foxt, He is Flaw’d, He is Fluster’d, He is Suttle, Cupshot, Cut in the Leg or Back=, He hath seen the French King, He hath swallowed an Hair or a Taven-Token, he hath whipt the Cat, He hath been at the Scriveners and learned to make Indentures, He hath bit his Grannam, or is bit by a Barn Weasel.

Also, from Cassell’s Dictionary of Slang:

Cut adjective [18th century and still in use] drunk (abbreviation for Cut In The Leg, a facetious euphemism for being staggering drunk).

(www.wordwizard.com)

And from The Social Historian, in the list of 17th century euphemisms for being drunk, you have:

To have cut your leg.

so the idea appears to be from the effect of alcohol on the way you walk, as if your legs were “cut” that is unable to support you.

(www.merriam-webster.com/images)