Dunaway was born in Bascom, Florida, the daughter of Grace April (née Smith), a housewife, and John MacDowell Dunaway, Jr., a career non-commissioned officer in the United States Army. She is of Scots-Irish, English, and German descent.
She attended the University of Florida, Florida State University, and Boston University, but graduated from the University of Florida in theater. In 1962, Dunaway joined the American National Theater and Academy.
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