No, he restricted himself to only Jesus Christ and Him crucified with the Corinthians. He did not restrict himself to this anywhere else. In other places Paul preached Jesus and Him crucified, and other things.
And we also know about church from Peter, James, and John, not just Paul.
99% of what we know about church practices, the do's and don'ts, came from the epistles of Paul. Peter, James, and John, did not teach us the construction of the church, that was given to Paul by God.
He preached the same at Corinth as he did anywhere. He peached that every promise from God comes through the Cross. Everything concerning the Chrsitian life is centered around the Cross. Without the Cross nothing is available for the Christian. We must have a solid foundation in the Cross of Christ (our salvation) or we have nothing.