OK, so after Truman wins, what happens next? How does Thurmond's career play out? What happens to the EC? What about civil rights?
I don't see any reason to believe that there will be too much fuss about the Electoral College going on. At the end of the day, there is a constitutional mechanism for dealing with situations like these, and they will play out as intended. Sure, some Republicans might murmur in bitterness, but no more than Democrats murmured in 2000 or 2016. They might have lost, but the election will widely be seen as a fluke, and the outcome as messy but within reason. It certainly wouldn't qualify for the term "Constitutional Crisis", and there'd be no great move for changing the basic rules by which presidential elections are fought.
As for Thurmond, I'm sure he can yet have a good career as Governor of South Carolina, and a popular one at that, and if he later on wishes to, he may very well run, and get elected to the Senate. By that point, enough time has passed that the other Southerners will welcome his as in OTL into the Southern caucus. Sure, Richard Russell, Jr. may shake his head and make wry comments about how Thurmond is a liability for the cause, but he certainly won't exclude him from the caucus. Any ally is a useful ally.
I see no reason to believe that this will have any tangible effects on the civil rights issue in the short term, Truman was bound to fail in that no matter what, and in the long term, the variables that need to be taken into account are so manifold that one cannot chart out a clear trajectory for where American will probably end up.
Now, Truman may end up having to chart a more careful path in his policies, both domestic and foreign, as he does not have the mandate of the comeback kid who defied all expectations and still secured his election. But I fear I can make no comments as to the specifics of this.