Very Few blacks were voting in the south before 1964.
There were poll taxes, and literacy tests for blacks that no person black or white could pass.
Prior to the voting rights act of 1964 there were next to no blacks voting in the south. Blacks could not sit in a white restuarant or vote in all the old confederacy states. The voting rights act of 1964 was passed to stop the Southern states from preventing blacks from voting. If blacks in the south could have voted before 1964,there would have been no need for the voting rights act of 1964.
The first inroads with blacks came in 1948 when Harry Truman integrated the military. That meant the military bases in the south. But Blacks could not vote in many northern states either.
But FDR got few if any black votes. Even in southern ohio it was very hard for blacks to vote.
When I was growing up in the 1950's the phrase used to denote you were in trouble was to say "You are going to be sweating it like an "N word" at an election." It was not considered healthy for black to stay over night let alone vote in Pike County Ohio. In 1966 I hired a black announcer at a radio station in north west ohio. The sherrif of the county told me that if I wanted to keep breathing I would fire the, as he phrased it, "damn n word".
It wasn't just south of the mason dixon line that blacks did not vote in the USA. In Chillicothe Ohio public schools were segregated and blacks were not permitted to use pubic restrooms, drink from public water fountains, or eat in restuarants until the late 50's. They had a hard time voting.
Blacks really became Democrats as the result of Nixon. Nixon completed the trade with LBJ in 1968 when the NIxon southern strategy traded white southernors (the solid south) for northern blacks.
It was Nixon's outreach to men like Strom Thurman that changed white southernors from Democrats to Republicans and Blacks from Republicans to Democrats.
Orval Faubus, George Wallace, Lester Maddox, and Strom Thurman were Democrats when they espoused racist policies. Klu Klux Klan wizzard Robert C. Byrd was the Democratic norm prior to 1964.
You are no doubt believe that Blacks voted for Klu Klux Klan members in large numbers. They did not