Not too hard, I can do it without nukes.
In 1933, the Business Plot is successful, they find a more right winged general to lead the march on Washington and they take down FDR. Prescott Bush is made dictator of the United States, now know as simply America.
In response to this, the heavily FDR supporting south attempts sucession again in 1934, but it fails and military forces occupy the deep south. After some obvious racism from the mostly white army, African Americans in the south launch a revolution in 1936 that suceeds and creates the Republic of Afrika, a communist nation where only the black can live.
Without FDR's reforms, the dustbowl continues and is much worse, and the midwest rises into a full revolution of pissed off farmers. The midwest of America becomes the Plains Republic and is a nation with a strong army but lots of poverty, some reforms are pulled off and in the mid 40's the dustbowl is over, but it leaves the nation scarred.
With the west now completely seperated from America, it falls into brief anarchy before an aristocrat takes control, the nation of California is born and becomes a direct democracy, it keeps good relations with the Plains Republic and is probably the best place to live ITTL.
Finally, Alaska, Hawaii, and US Territories notices at various points in this timeline that the fascists in Washington can't do anything about them seceeding form the United States, and become weak governments all around.
And there we go, Balkanized America, after 1900, no nukes.