You ask: In both of the sentences above, why can't we say "....might have..."?
I'll answer: Who says you can't? Let's try it, and see what happens:
“I thought she had a good chance to get on the medal stand,” Dunn said Monday. “Andif she got a good jump, I thought she might have a chance to win it".
That version seems grammatical, too.
Essentially, the speaker is saying:
If she had a good jump, she might have won.
There are many ways to say that, and you may not hear the most straightforward way of saying it the moment a microphone is thrust into a coach's face. So, the coach says,
If she got a good jump, I thought she might have had a chance to win [the race].
In this original, the words "if...", "I thought...", "might have...", and "a chance...," we are pretty well-nested in conjecture and conditional language. The result is a sentence where the word had could be omitted or included, and the sentence pretty much means the same thing, especially if you change the punctuation around:
"With a good jump," I thought, "She might have a chance to win the race."
The same holds for your first example:
Eddie didn’t talk much about it, but I thought she might have had a problem with alcohol.
Eddie didn’t talk much about it, but I thought she might have a problem with alcohol.
The phrase "I thought she might" puts the sentence in the past tense, so the had can be removed. If I remove that phrase, though, the had needs to stay in place:
Eddie didn’t talk much about it, but she had a problem with alcohol.
Remember, we are dealing with spoken words that were written down, not an author's polished work. A writer may have revised the sentence to read something more like this:
Eddie didn’t talk much about it, but I thought she had a drinking problem.
Once again, after the conditional language has been removed, the had is required. Also, just like last time, I can remove the had, particularly with a change in punctuation:
Eddie didn’t talk much about it, but I thought, "She might have a problem with alcohol."
Best Answer
The two marked sentences are both direct "speech", although Oates does not mark them with quotation marks—probably because they are not spoken but only thought. The must here therefore represents present tense.
But the past tense of must is not had to, though had to may substitute for it. The past tense of must is must†.
In any case, there is no justification for using either may or might here; either would completely change the meaning.
† Historically, in fact, must was the past tense of the verb mote, which long since disappeared from use. Must appears to be following in its footsteps, being more and more replaced by the more flexible quasi-modal HAVE to.