Yes you are correct, Although it was feared at the time the Democratic party would split over the 1957 legislation, 1964 legislation did split not destroy the Democratic party. Liberals stayed democratic and the racists conservatives went to the Republican party. Think Strom Thurmond, Jesse Helms.
But what happened to the Republican Party on racism? It is two very different institutions pre 1960 and post 1960. They abandoned Lincoln for Goldwater. Employed the southern strategy and gave up on the black vote to create a white majority conservative party.