I can relate to much of this.
I grew up in the South (though not rural or Appalachia) and at age 10, I made the conscientious decision to get rid of my southern accent. It sounded "unintelligent" to me but more than that, it seemed to bely my moral values. As judgmental as this sounds (and it's true, it is judgmental), I looked around at the values that are often attached to the South -- bigotry and racism being the big two -- and said, "no."
So, at age 10, I lost my accent. I first stopped saying "y'all" -- but it quickly morphed into my entire accent changing. My parents/sister and most of my friends still had an accent, but I lost mine. With the exception of a few words or phrases, I don't even say southern colloquialisms. (I do still call every type of soda "Coke" -- but that's just Atlanta pride).
When I take those tests that determine dialect, I never get the South, I almost always get midwest or Southern California. (That was the goal, so go me?) Even drunk, I don't slip back to a southern accent, simply because it doesn't exist as part of me anymore.
The fact that I made that decision at age 10 -- and to be VERY clear, it was my own decision and not one pushed/encouraged by my parents (in fact, they probably rolled their eyes) -- I think says a lot about just how palpable those stereotypes that you mention are. Because at that age, when given the opportunity to lose an accent (I had speech impediment, was taking speech therapy a few times a week and had to basically re-train myself to talk. I decided to hack the system and get the accent I wanted in the process), I took it. I'll be totally honest, I don't regret it.
But still, I do often think about the code switching that I chose to take on at that age and what that means. For my career now, which often includes going on television -- it's realistically probably served me well to have no accent. Even in high school working in retail, it often served me well to "not sound like I was from the South" when selling someone a $1500 digital video camera or charging $50 an hour to repair a computer.
Still, I do think about it a lot. I think about why it was so important, even at that age, for me to reject something like my voice/accent.
My husband (a software engineer) is a lot like you in that he is able to switch his accent on and off depending on who he talks to. When speaking with his parents or grandmother, his Tennessee twang comes out. With colleagues or people in our neighborhood, there is a slight hint of an accent, but nothing major. And he does it for the same reasons.
No one wants to sound like a redneck in a professional setting, especially one stereotypically white-collar like software engineering. And yet, we don't talk about why that is the case -- or why where we are from matters.