I was born in Houston, Texas and came to California when I was 2 years old. I have been around the California dialect all my life, but, was raised by parents who grew up in Texas.
I never had any idea that I had any sort of southern accent until I went to acting school in my late teens. They had us perform monolougs in the dark and audio taped them. The teacher said to me, "Joe, you say git for get, kin for can, jest for just, yuh for you and fer instead of for".
I disagreed, and she played one of my lines back:
(the phone rings)
Me: Kin yuh git that fer me?
I have almost the same story, except I was born in California to parents who came here from Minnesota. I never knew I had an accent 'til I met a college linguistics professor who detected it.