Dunagan was born in Monahans, Texas. Her father was a Coca-Cola bottler and president of the Texas Historical Association. In an interview Dunagan said, "My heritage is a long line of Southern Baptist and Methodist preachers—who were all just frustrated actors." Dunagan got a degree in music education and married young, but her marriage did not last long. Following her divorce, Dunagan's parents agreed to fund graduate studies at the Dallas Theater Center. While writing her Master's thesis Dunagan lived in Mexico and got engaged to a bullfighter. Dunagan never married the bullfighter, who moved to Spain, and went on to pursue her acting career in the United States.
Dunagan began her acting career performing in regional theaters such as the Asolo Theatre in Florida and the Actors Theatre of Louisville before trying her luck in New York City. She made her Broadway debut in the 1979 production of George Bernard Shaw's Man and Superman at Circle in the Square as an understudy for Ann Sachs. Sachs became ill at one point during the show's run and Dunagan filled in for her with great success. This exposure led to her to being signed with International Creative Management and helped forward her career.
In 1981, Dunagan performed in the first national tour of Children of a Lesser God. While on this tour, Dunagan visited the city of Chicago for the first time and immidiately fell in love with the city. After the end of the tour, Dunagan moved to Chicago and has lived there ever since, performing in more than 30 theaters in the Chicago area. Her work on the Chicago stage has garnered her three Joseph Jefferson Awards and three After Dark Awards.
Dunagan has also periodically worked in films including, The Naked Face, Running Scared, Men Don't Leave, Losing Isaiah, and Dimension. She has also appeared in more than ten made for television movies, in the tv mini-series A Will of Their Own and Amerika, and as a guest star on the television shows Prison Break, What About Joan, and Missing Persons.
In 2007, Dunagan returned to Broadway in the Steppenwolf Theatre's transplant production of Tracy Letts' August: Osage County. For her performance, Dunagan won a Tony Award, Theatre World Award and a Drama Desk Award.
