If Oedipus had accepted the Crown of Corinth and ruled both cities simultaneously, what different direction might Sophocles' play Oedipus Rex have...

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.