Green’s Dictionary of Slang

rummed up adj.

tipsy, drunk.

1902		Indianapolis Jrnl 25 May 15/2: Neither [...] does the ruymmed-up tar bother [...] over the stripes and stars on the sleeve ciffs of the officers.
1928		M. Prenner ‘Sl. Synonyms for “Drunk”’ in AS IV:2 102: ‘Boozed,’ ‘lit,’ and ‘rummed’ are usually followed by up.
1938		D. Runyon ‘Lillian’ in Runyon on Broadway (1954) 292: She is good and rummed up.
1944		D. Runyon Runyon à la Carte 63: Rudolph staggers a little now and then like a guy who is rummed up.