Does Oedipus resolve his own dilemma of having too much pride in Sophocles' work, Oedipus Rex?
This is a great question. Here are a few points to consider.
First, as you state the tragic flaw of Oedipus is his pride. In fact, I would go as far as to say that the whole play is driven by Oedipus' pride. In light of this, it might seem that Oedipus does not resolve his pride issue. However, there are a few indications to show that Oedipus finally does realize his pride.
First, by the end of the play, Oedipus definitively knows that he is the one who committed the crimes. In light of this, he is forced to deal with his pride. There is simply nothing that he can do. More importantly, he accepts his fate. We can see this even more clearly in the two other Theban plays. In the final play, he becomes a blessing for Athens. He is now a wise man.
Second, there is a motif in the play of blindness. In the beginning of the play he see physically, but he cannot see figuratively. We can say that he is blind to his pride. By the end, he blinds himself physically, but he is now able to see much more on a spiritual level. This suggests that he does deal with his problem of pride.