If Oedipus had accepted the Crown of Corinth and ruled both cities simultaneously, what different direction might Sophocles' play Oedipus Rex have taken?
It actually would have been logistically impossible for Oedipus to have become king of both Corinth and Thebes. One reason is because had he returned straight home after consulting the oracle, he never would have traveled to Thebes in the first place. Had he not been heading in the direction of Thebes, he would have never encountered on the same road the man that turned out to be his father, nor would he have killed that man. The prophecy was only fulfilled because Oedipus tried to run away from his fears.
According to the story Oedipus relays, when he was a young man he traveled to the oracle at Delphi to learn the truth about his parentage after hearing a drunken man at a banquet proclaim that Polybus was not his real father. However, Oedipus was only told by the oracle that he would one day kill his own father and sleep with his own mother. It was at this point that Oedipus decided to run away from Corinth with the intention of never seeing the prophecy fulfilled, as we see in Oedipus's lines:
I heard and fled, henceforth to share with Corinth only the stars, where I would never see completed the disgrace of those evil oracles of mine. (822-825)
However, Oedipus happened to be at a place where the three roads leading from Corinth, Thebes, and Delphi meet. Therefore, instead of returning to Corinth, Oedipus continues on to Thebes instead. At that same moment King Laius was traveling from Thebes towards Delphi to consult the oracle himself. He and his traveling companions ran Oedipus off the road when they saw him, and in his state of anger over the oracle, plus his fatal character flaw of excessive pride, Oedipus struck out at the man and all his traveling companions, killing them all. Therefore, had Oedipus never decided to try and run away from his fate and his fears by running away from whom he thought was his father, he never would have headed towards Thebes and never would have encountered his real father. Oedipus would have been crowned king of Corinth and lived to never lay eyes on Thebes nor any Thebans, including his mother.