Green’s Dictionary of Slang

overshot adj.

tipsy, drunk.

1605		Chapman & Jonson Eastward Ho! IV i: Death! Colonel, I knew you were overshot.
1931		T.R.G. Lyell Sl., Phrase and Idiom 668: There are innumerable synonyms applicable to different degrees of intoxication. Those most in general use are [...] muddled, overcome, overshot, [etc.] [OED].
1942		Berrey & Van den Bark Amer. Thes. Sl.