There was never any organized "Dixiecrat" Party. Strom Thurmond in 1948 usurped the Democrat ballot position in the states he carried. Thurmond's followers made no attempt to compete with the Democrats for any state or local offices, including U.S. Senator and Representative.
The white supremacists in the South exercised their control through the Democratic party organizations in their states. With the Federal interventions of the 1960s, their control was cracked - but bolting the party would have shattered it. And they had way too much invested in the political situation - they held lots of offices, controlled appropriations and agencies which had nothing to do with race. Why burn all that down for a quixotic fight they couldn't win?
Also recall that by 1972, much of the South was already voting Republican at least some of the time. There had been Republican governors in Florida and Arkansas, Republican Senators from Tennessee and Texas.
Some facts:
- Republicans were never completely shut out of the South. Even at the height of Jim Crow, Tennessee was a swing state, electing a Republican governor in 1910, 1912, and 1920. A Republican was elected to the House from Texas all through the 1920s.
- The single-minded attachment of Southerners to the Democrats was an unnatural alliance. Outside the South, the Democrats were the party of big-city machines, immigrants, Catholics, unions, and liberals. The South was rural, devoid of immigrants, anti-Catholic, anti-union, and conservative. What created and sustained that alliance was their mutual opposition to the Republicans. The Jim Crow regime was maintained by force: black disfranchisement enforced by Klan terror including lynchings. Republicans made a few feeble efforts to break up the Jim Crow regime, but these were blocked by Southern Democrats in Congress supported by Northern Democrats. When Northern Democrats ended that alliance in 1948, the reason for absolute Southern loyalty to the Democrats ended. After that, it was inevitable that the South would revert toward its natural state, with both parties being competitive.
- In 1972, the Democrats turned sharply to the left. In 1968, there was a left insurgency against the Democrat mainstream, which was turned back at the convention with much bitterness. In 1972, the insurgents took control. Note the position of Chicago Mayor Richard Daley. In 1968, Chicago police drove left-wing demonstrators off the streets, and IIRC Daley helped silence dissenters on the floor. In 1972, he and the rest of the Illinois delegation were removed from the convention for being insufficiently "diverse" (to use the current coinage). The previous six years had seen a wave of cultural radicalism, accompanied by political radicalism. The drug culture. "Free love." "Off the pigs." The "anti-war" movement's attacks on the honor of the U.S. armed forces (Southerners are very pro-armed forces). In 1972, it appeared that these forces were now in command of the Democrats. This was an exaggeration, but a useful one, as Republicans labelled the Democrats the party of "Acid*, amnesty**, and abortion." (* "Acid" was a slang term for LSD. ** "Amnesty" for Vietnam-era draft evaders and deserters.)
Nixon's "Southern strategy" was to attack the national Democrats on these grounds, say nothing about race, and refrain from challenging Southern Democrat incumbents down the ticket. It allowed him to sweep the South in 1972, though it gained almost nothing for Republicans downticket.