The story
From dead + beat. As an adjective, of a person, to be exhausted, first use appears c. the 1820s. During the American Civil War, it also became a derogatory adjective, in the sense of "a person that defaults on their debts or avoids responsibilities" and "sponger", "vagabond". This sense dates from c. 1863. Possibly related to good for nothing.
Etymology from Wiktionary, CC BY-SA 4.0.