Adžić came through the youth system of Red Star Belgrade, making his senior debut on 2 December 1989, at the age of 16.
He came on as a substitute for Darko Pančev in a 3–1 away league win over Spartak Subotica.
In the first half of the 1991–92 season, Adžić played on loan for Borac Banja Luka, before returning to his parent club.
After hanging up his boots, Adžić served as assistant manager to Ratko Dostanić and Walter Zenga at Red Star Belgrade from 2004 to 2006.
He also spent one season as manager of Montenegrin club Rudar Pljevlja.