If you do that magic thing called research, you'll find that roughly 1% of the "Dixiecrats" switched to the Republican party. Like, seriously, a dozen of them. The rest did not.
For every Storm Thurmond, there were 10 Robert Byrds.
The Democrat party was in shambles. The Republican party was not.
You'll also find that a lot of citizen / voters moved from the north states to the south during that time for economic reasons, and to escape the growing socialism of the major northern cities that began in previous years. McCarthy went off the deep end, but his perceived enemy was quite real.