Some of those Dixiecrats were hardly conservative, just old-school New Deal liberals who happened to be against civil rights.
As for Harry Truman and JFK-style liberals, many of their goals have already been achieved, so it's hard to say what their old-time liberalism would mean today.
When people say that JFK would be a conservative or a Republican if he were around today (which I don't really think, if only because it would mean bucking the rest of the family), what would that make Webb?
Some were pro-New Deal, some were against it. On the other hand, most Dixiecrats were solid supporters of state's rights and right to work laws.