He almost got away with it. His plan was to fly to Rhodesia where he believed he would be treated as a hero by the Apartheid gov't of Ian Smith (in truth, they would've likely extradited Ray back to the United States).
Excellent page-turner of a book about the King assassination and ensuing manhunt:
went to Atlanta and then to Canada and England before being arrested in July 1968.
He was actually arrested in June of 1968. I recall the news breaking during the televised train trip of Robert Kennedys body preceding his funeral