The Blacks political problem doesn't help the situation. They're extremely politically polarized. If 95%+ vote democrat that means there will never be any internal competition in running their homelands which results pragmatically and historically in ridiculously incompetent small governmental bodies which results in failure. Meanwhile on the larger stage minorities can be kingmakers if and only if they are not polarized, polarized equals marginalized. The D party knows they'll get 95%+ of the vote regardless what they do or promise so they have no motivation to do or promise anything for Blacks. Meanwhile the R party knows they'll get rounding down to 0% of the votes no matter what they do or promise so they also do nothing for Blacks. Net result is Blacks will never have any political influence on national politics until their voting records are closer to 50:50, or at least further from 100:0, until that happens they are disenfranchised above the local level.
That is a side cultural issue in that no one wants to live in a dysfunctional local government if they can avoid it, and disenfranchisement at a higher government level means their unique concerns are going to be eternally ignored if not actively worked against, and neither trend helps integration overall.
The situation in the USA is somewhat similar to the townships system in the bad old days of South Africa, just unofficially. That system wasn't really conducive to integration or positive race relations in general.
This was by design: read up on the "Southern Strategy" if you'd like to get a better understanding the recent history between the GOP and African Americans. The Republicans found it politically expedient to oppose the civil rights movement, yet before this, the Democratic party was the party of the KKK.
One small correction: Blacks are not the majority in any state. Mississippi has the highest percentage, at 37%, but it is still 59% white.