Down here in Texas, which is supposed to have a heavy accent, I find that most of the time we talk normally to each other, at least here in the big city of Houston. It's mostly mexicans and country people that have accents they can't shake if they want to.
I think you get used to your surroundings and don't notice the accent. I worked in a tourist trap here in Colorado one summer and found myself speaking "Texan" at times because that was the most prevalent accent. I can still distinguish other "Southern" from "Texan" most of the time.
I grew up just west of Houston in Richmond - where the Old Three Hundred landed...most of those guys were from the Deep South...there is a definite Southern inflection in the old timers here. But I learned real quick how to distinguish an East Texan from a West Texan...President Bush has a definite West Texan accent, whereas most of my husbands relatives have what I call 'cat-scratch East Texan' But my parents came from Central Texas and whenever we traveled out of state, people would be surprised to learn we were from Texas. "You don't sound like it." they would say.
That, and I grew up being involved in theatre, and you KNOW they do everything they can to get rid of ANY accent.