mid-15c., "lax, flaccid, soft, tender" (obsolete or dialectal), from Old French lasche "soft, loose, slack, negligent, cowardly," from laschier "loosen," from Late Latin laxicare "become shaky," related to Latin laxare "loosen," from laxus "loose" (from PIE root *sleg- "be slack, be languid"). The main modern sense of the word, with reference to plant life, "luxuriant in growth," is first attested c. 1600, in Shakespeare. Related: Lushly; lushness.
"drunkard," 1890, from earlier slang meaning "liquor" (1790, especially in phrase lush ken "alehouse"), of obscure origin; perhaps a humorous use of lush (adj.) or from a word in Romany or Shelta (tinkers' jargon). It also was a verb, "to drink heavily" (1811).
LUSHEY. Drunk. The rolling kiddeys had a spree, and got bloody lushey; the dashing lads went on a party of pleasure, and got very drunk. ["Dictionary of Buckish Slang, University Wit, and Pickpocket Eloquence," London, 1811]
Hence also Lushington humorous generic name for a tippler (1823). It was an actual surname.