That's the reality as opposed to the DemonRAT myth. But it wasn't always that way as this map so vividly illustrates:
In the South, less than a century ago, what's now the home of the Conservative base repudiated Calvin Coolidge, a Patriot admired by President Reagan, and embraced a leftwing 'RAT, John Davis. The story was the same in 1928 when far-left zealot Al Smith was supported by the Deep South rather than Herbert Hoover (a RINO but still more Conservative than Smith):
That pattern remained in place when FDR swept the South in the next election cycle:
And so the song remained the same for years thereafter with the South being the liberal part of the Republic and, oddly, most of New England being Conservative historically. As you stated, it took the dying of the old guard 'RATs in Dixie to make the transformation to the Conservative bulwark we see today. Conversely up in Yankee-land (notably Coolidge's home state of Vermont, now a socialist cesspool), it was the dying off of traditional Conservatives that led to the regions decline into leftism.
If you're still with me after this lengthy post, my contention is that the parties never switched. The party of Lincoln is the same today (modulo the RINO element) and the 'RAT party of leftwing radical William Jennings Bryan (he the prime force for the unconstitutional income tax) is unaltered from what it was in the 19th Century. Instead, it was the people who switched, through attrition and via migration between the regions. As the modern South emerged, Conservatives from other parts of the country found the climate (both the weather and politically) to their liking and brought their pro-American Patriotism to Dixie! Well, at least that's how I see it.