Another way to search for info about this in by googling

alcoholics, rheumy eyes

No quotation marks.

I've never heard the term 'alky eyes'. The word 'alky' used in reference to people who drank lots ('never knew he drank until I saw him sober once' kinda folks) was in usage when I was a kid, but seems to have disappeared when alcoholism was seen more as a disease than a social failing or character 'flaw'.