The two books of Corinthians were written by Paul around AD 55, from Ephesus and Macedonia. This means they were written before Romans. The focus of these letters is pastoral rather than theological -- Paul is addressing problems that have arisen in the church. The Corinthian church has gotten "a bad rap" over the years as a disorderly and licentious group. But this was to be expected in a city on the Greek mainland, with its pagan religious background of polytheism and prostitution.
We might consider Corinth and Jerusalem to be opposite poles of spirituality -- antinomianism vs legalism. The Gospel as preached by Paul was not trying to side with one or the other, nor take a middle ground. Instead, the Gospel differentiated itself from both rivals -- it challenged the Greeks to "set your minds on things above, not on earthly things" (Col 3:2). To the Jews, it offered moral freedom not found in the Law. Christianity produced a spiritual transformation in converts from either background. Corinthians shows us that some Greeks had as much difficulty with forsaking their past traditions as some Jews did in Acts.
There is not much direct racial teaching in these books. Unlike Romans, Paul is not trying to solve problems of racial/religious history involving God's sovereignty and human destiny. But we do see the same priority given to unity in the body and the life of the Spirit as we do in Romans.