Except the switch didn’t happen until decades later. Virtually no Dixiecrat switched parties. And Carter won every southern state except Virginia in 1976. He did better in Alabama than in New York. Reagan and George H. W. Bush muddy the analysis because they won everywhere, including deep blue states. But Reagan did better in wealthier, more educated southern metro areas in the south than in presumably more racist rural areas. Clinton won a number of southern states in the 1990s. Republicans didn’t win a majority of southern congressional seats until 1994.
Apart from it making no sense to say Dixiecrats protested Democratic support for the civil rights act by switching to the party that not only supported it more, but supported all the previous civil rights acts, that narrative ignores the actual issues that mattered to voters in the 1980s and 1990s. Abortion, foreign policy (patriotism), and economic policy became defining issues during that time. And on all those fronts, ”new south” voters were more aligned with the Republican position.
Put differently, democrats support for segregation was keeping southern democrats in the party. It was preventing what would otherwise be ideological sorting of a voting bloc that was already more religious conservative, patriotic, and was benefiting economically from deregulation as industry moved from northern states to southern states.
> Many of these voters continued to oppose racial (and "legal") equality for many years, and in some cases are still do.
Nobody opposes legal equality. Many republicans oppose what we now call “equity”—taking race into account to produce racially even results. If you look at “racial resentment” tests, for example, the fundamental difference in attitudes that appears is that liberals have a special sympathy toward black people, while conservatives have equally unsympathetic attitudes towards everyone: https://scholar.harvard.edu/files/renos/files/carneyenos.pdf.
Your Alabama example illustrates the point. Alabama is trying to draw a map favorable to republicans for the same reason Maryland draws a map favorable to democrats: https://www.brennancenter.org/our-work/research-reports/mary.... Alabama isn’t trying to disadvantage black voters, just as Maryland isn’t trying to disadvantage white voters.