Yes, it’s a myth. Many Southern Democrats (sometimes pejoratively referred to as Dixiecrats) were actually more fiscally conservative and anti-federal power than the progressive Northern Democrats who believed in a stronger federal government. But that’s where their similarities to conservative Republicans end. The Southern Democrats were also proud segregationists and had been since Reconstruction. The Southern Democrats were always the more powerful part of the Democrat Party for decades even though the Republicans were far more prevalent after the Civil War.
Republicans, however, had been both fiscally conservative as well as socially conservative. They were the anti-slavery, anti-segregation party of Lincoln; although they did have bouts of progressivism at the turn of 20th century with likes of Teddy Roosevelt.
However, when the progressive wing of the Democrat Party started to outweigh the Southerners after World War II and gain more and more power, things started to shift. It was during the end of the Civil Rights era that they finally lost all their power. At the same time, more rigid conservatives like Barry Goldwater were also trying to shift the power within the Republican Party from the more liberal (sometimes referred to as “moderate”) establishment Republicans towards the more libertarian-conservative base. Goldwater attempted a “southern strategy” in which he appealed to disenfranchised conservative voters in the southern states who aligned to the conservative notion of states rights (not to be confused with an appeal to segregation as Democrats like to suggest.) What cost Goldwater the election was his opposition to the Civil Rights Act, which LBJ (his opponent) had signed into law. LBJ used this to his advantage to smear Goldwater unfairly because Goldwater did not oppose the Civil Rights Act for racial reasons but rather because he found this rendition of the bill to be a detriment to certain liberties. Unfortunately, the smear campaign was successful nonetheless. Even further tragic was how Al Gore Sr. was able to get reelected despite his opposition to the Civil Rights Act for more segregationist positions.
Nixon and Reagan also applied a similar southern strategy to appeal to more conservative citizens in these states. This is where most liberals come up with the notion that the “parties switched.” The truth was that segregationist George Wallace left the split the Democrat Party and still won the southern states from Nixon in the ‘68 election. It wasn’t until 1980, when Ronald Reagan ran for President, where the Republicans were able to get a majority of the southern states. By then, most segregation views had dissipated into the past and were only held by nationalists and other fringe groups.