Merriam-Webster defines high as a synonym for drunk, along with more than a dozen other synonyms. I have never heard of high being used to convey a lesser degree of intoxication than drunk. To convey some stage of a slight effect from alcohol, the word tipsy is often used.
High is also commonly used to mean intoxicated on drugs, or in a sense of intense pleasure comparable to intoxication, i.e, "Sunshine almost always gets me high...." lyrics from John Denver circa 1971.
Edited to add another example: The Kingston Trio had a hit from 1958, "Scotch and Soda", which has been widely covered including by Manhattan Transfer in 2011. Lyrics are:
"Scotch and soda, mud in your eye, baby do I feel high, oh me oh my, do I feel high.
Dry martini, jigger of gin, oh what a spell you've got me in, oh my, do I feel high.
People won't believe me, they'll think that I'm just braggin', that I could feel the way I do, and still be on the wagon.
All I need is one of your smiles, sunshine of your eyes oh me oh my, do I feel higher than a kite can fly!
Give me lovin', baby, I feel high."