It's important to recognize that the Dixiecrats , though they ran as "Democrats," voted with the Republicans on substantive issues. And this was not just in the Senate, it was in both Houses. They were, in effect, Republicans dressed up in donkey uniforms. Those few that didn't switch parties didn't "die out", rather they eventually got defeated at the polls, as more and more White Southerners switched parties. I'll submit Mississippi's Gene Taylor as an example of what I am referring to. Taylor ran for the U.S. House and was elected multiple times as a "Democrat" but nevertheless had a virtually 100% Republican voting record on substantive matters. He didn't switch parties. Although he was a Republican in every respect, except for the label he adopted, he was defeated when he was opposed by Steven Palazzo running under the Republican Banner in 2010. Taylor later did switch parties, but it was too late to resuscitate his political career.
Lyndon wasn't giving up much when he said he had handed the South to the Republicans, because the Southern Democrats were never supportive of the Democrats' progressive legislation.
This strange reversal of political labels goes back to the civil war. After Southern blacks were finally able to vote in significant numbers they voted for Republicans, the Party of Lincoln, but practically no Southern Republicans running against Dixiecrats prevailed at the polls. This was largely due to poor turnout among Southern Blacks who were intimidated at the polls and legally prevented from voting by poll taxes and other, then legal, measures adopted by Southern Whites who controlled the Deep South state legislatures. This changed with the passing of Federal Voting Rights legislation (recently gutted by the Supreme Court). A decade after the Voting Rights Act was passed a few Dixiecrats began to come out of the closet as Republicans. It was as though a dam had broken. Within a few years, one after the other Dixiecrat jumped ship. And white voters quickly switched their allegiance. The educated Black voters simultaneously began voting for Democrat candidates but some black voters continued to identify with the party of Lincoln, apparently not recognizing that they were voting against their best interests. Today, however, Southern Blacks are solid supporters of Democrat candidates. Going forward, with black voter turnout increasing and growing progressive support among Independent white voters, the Republican Party can no longer count on the Deep South remaining solidly Red without Gerrymandered districts and restrictive voter laws.