Did the use of the term lush for a drunkard originate with someone named Lush?

Yes, Dr. Thomas Lushington (1590-1661), an English chaplain who liked his liquor.

The City of Lushington, a London drinking club, may have borrowed its name from him.

In the nineteenth century, the City of Lushington in turn became the source of the word lush, originally a slang term for beer and eventually another word for drunkard.