I was also raised in the SB church in TN, during the Al Gore, Sr. era. Although I've pulled away from the church in my later years, I'm still fond of those days and the old-fashion country fellowship.
All the churches I recall from back then were fairly consistent in teachings and attitudes, and I never recall politics mentions once in my years there. From what I hear from folks back there, things have changed a lot, but maybe not in all the churches. After the Southern strategy of the Nixon years, the entire political climate began to shift, and now it's almost all red.
Honestly, I was never aware of the autonomy from the SBC you mentioned, but it wasn't discussed much back then. My single foray back to church here in KY was back in the early 90's, and unfortunately I had a bad experience with that one. Perhaps I should have shopped around more.
As a side note, I have the highest respect for Jimmy Carter and the work he and Rosalynn are still doing. What an inspiration! 
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