1560s, "go this way and that in speech or action," a sense now obsolete; from 1680s as "start suddenly aside, shift suddenly," as to evade a blow;" 1704 as "to move to and fro, shift about;" origin and sense evolution obscure. Perhaps it is from or akin to Scottish and Northern English dodd "to jog" (1570s).
Transitive sense of "to evade (something) by a sudden shift of place" is by 1670s. It is attested from 1570s, and common from early 18c., in the figurative sense of "to swindle, to play shifting tricks (with)." Photography sense of "use artifice to improve a print" is by 1883. Related: Dodged; dodging.
Dodge City, Kansas, was laid out in 1872 and named for U.S. military man Richard I. Dodge, then commander of the nearby army fort. It later was notorious in Wild West lore as the home of Wyatt Earp and Bat Masterson.