Hiya!

Yes, realignment from the democrats to the republicans in the southern United States throughout the 1950s and 60s definitely occurred and is easily observable. This is known academically as the realignment of the South. The southern states were, prior to the civil rights movement, almost completely dominated by the southern democrats. Starting in 1948 and continuing throughout the 50s, the Republican party began to gain influence in the region, and through the course of the civil rights movement, began to dominate it. What's important to recognise is that neither the southern democrats of 1948 nor their contemporary Republican opponents bore much resemblance to their respective party positions today.

Seth Mckee, in his essay The Past, Present and Future of Southern Politics outlines the critical factor catalysing the Southern Realignment - the democratic backing of civil rights, namely at the 1964 election, and the republican opposition to the same:

" Like Truman sixteen years earlier, President Lyndon Johnson understood that a pro–Civil Rights position was a winner in presidential politics. Johnson spearheaded and signed the 1964 Civil Rights Act, which banned racial discrimination in all places of public accommodation, and he routed Goldwater in the 1964 election—with Goldwater winning just six states: the entire Deep South and his home state of Arizona.

"After 1948, the Republican Party gained a toehold in southern politics, soon becoming a substantial presence in presidential elections and making albeit sluggish but detectable progress in down- ballot races (mainly U.S. House contests). In the 1952 and 1956 presidential elections, the Democratic and Republican presidential candidates did not emphasize race, and, in the short-term, it was essentially dormant. But this would change abruptly with the efforts to desegregate Little Rock Central High School in 1957, a response to the Supreme Court ruling in Brown v. Board of Education (1954), which finally overturned the 1896 Plessy decision by declaring that separation of the races was inherently unequal.

"The numerous civil rights demonstrations of the early to mid-1960s culminated politically in the 1964 presidential election—a contest that solidified the distinction between the national parties on the race issue. Like Truman sixteen years earlier, President Lyndon Johnson understood that a pro–Civil Rights position was a winner in presidential politics. Johnson spearheaded passage of the 1964 Civil Rights Act, which banned racial discrimination in all places of public accommodation. This action proved fateful, because his Republican opponent, U.S. Senator Barry Goldwater, voted against the legislation and thus sent a clear message to southern whites that the GO P was the more conservative party on civil rights. "

To the socially conservative and largely anti-civil-rights south, this split was decisive, and resulted in a vast swing of support from the southern democrats to the republicans - and the exodus of many conservative southern democratic congressmen to the Republican party. Mckee's essay features several graphs which excellently plot the progress of the Southern Realignment.

To answer your second question - to what extent - the answer is that the southern re-alignment, by the time of its completion, was near total. The south, once dominantly democratic, was (and is) now dominantly republican in its voting habits. The southern realignment has been one of the key areas of study for scholars interested in the polarisation of US domestic politics throughout the second half of the 20th century, and its effects continued to be felt today.

Source: Mckee, Seth. "The Past, Present, and Future of Southern Politics." Southern Cultures 18 (2012) 95-117.

(Multiple edits for expansion and formatting.)