No you haven't seen any Democrats claim that the party switched names and that didn't happen. What did happen was the movement of the white South away from the Democratic party beginning in the 1960s.
After the Civil War the country settled into the modern two party system where elections were between Republicans and Democrats. Since the Republicans had lead the Union to victory in the Civil War, destroying the Confederacy, ending slavery, and killing hundreds of thousands of Southerners, most white Southerners treated the party like the plague. After the end of Reconstruction, white Southerners forced black Southerners out of the electorate and the region became a solid Democratic stronghold. This continued even as the Democratic party diversified in the 20th century, gaining a large liberal wing based in the North and even gaining the majority of black politicians.
In the 1960s, under President Lyndon Johnson, the Democratic party remade itself as a party of civil rights, passing the landmark Voting Rights and Civil Rights acts which ended Jim Crow and explicitly outlawed racial discrimination. White Southerners, who had loyally backed the Democratic party felt betrayed and began to abandon them. This happened pretty quickly in presidential races. Johnson's Vice President Hubert Humphrey (a Northerner who had pushed civil rights) did well in the South in 1968 (losing several states to segregationist third party candidate George Wallace), but after that the Republicans had a hammerlock on the region with Democrats only making significant inroads in 1976 and the 1990s when they ran white Southern Governors at the top of the ticket. The change was much more gradual below the presidential level. Democrats continued to hold significant seats in the South, Congressmen, Senators, Governors, into the 21st century. But those seats, especially ones at the state level, became increasingly rare. By the turn of the turn of the 21st century the white Southern Democrat was all but extinct. The Democratic politicians who remained in the South (outside of Florida which is a special case) were almost all either black, or in Texas Hispanic, and represented enclaves where minorities predominated. Southern voting has become very racialized, with whites voting overwhelmingly Republican and blacks voting overwhelmingly Democratic. Democrats who want to win statewide races in the South have been largely out of luck. Even the exceptions prove the rule. Democrats have done well in Virginia, for instance, where they control all statewide offices and a majority of House seats. But this is because of changing demographics, specifically a rising black population and the influx of more non Southern whites who hold more liberal views and depend on Washington DC or other upscale industries for their income.