Galveston County District Clerk Jason Murray offered the paperwork Monday. Murray says his situation has become a distraction that's unfair to people at the courthouse. He plans to leave within about two weeks.
Prosecutor Jack Roady has said he was prepared to seek removal of Murray, who was elected in November 2010 to a four-year term.
Murray in December pleaded guilty to assaulting his wife after a party in Houston. He received two years of probation.
Murray was arrested March 27 in League City on a charge of driving under the influence.
He has also been accused of sexual harassment of a co-worker.
Attorneys for Murray declined comment.