It wasn’t just “if you like your doctor you can keep your doctor” that was a lie.
Just about everything that Obama promised about Obamacare turned out to be a lie.
There was gradual voter realignment after the 1960s, not an overnight party position swap. Democrats controlled the Jim Crow South and most opposition to the 1964 Civil Rights Act; Republicans supported it at higher rates (~80% vs ~60-70%). Coalitions and platforms evolved over decades as Southern whites shifted GOP and Black voters solidified Democratic. Cruz correctly cites the historical record.