Ms Odinet announced her resignation on 31 December after spending time on unpaid leave.
In a letter directed to the Louisiana Supreme Court, Ms Odinet said she took “full responsibility for the hurtful words” that sparked the backlash.
According to local news broadcaster KLFY, the video was taken in her home, and shows her responding to a car break-in attempt. A woman’s voice can be heard saying “we have a n*****. It’s a n*****, like a roach.”
Ms Odinet initially said that her use of the racial slurs was the result of sedatives affecting her “fragile” mental state.
“I was given a sedative at the time of the time of the video,” she said in a CNN interview. “I have zero recollection of the video and the disturbing language used during it.”