Modal Verbs and Past Tense – Correct Form: ‘My Dog Could [Have] Jump Very High’

I believe that you could replace every were able to in your examples with could, but it might not mean what you think it means, and it could sound strange or misleading enough that some grammar books make it a rule that you must not use it. For example,

11.12.3 Specific achievement in the past
Could cannot normally be used when we are describing the successful completion of a specific action: was/were able to, managed to or succeeded in + -ing must be used instead.

In the end they were able to rescue the cat on the roof.
In the end they managed to rescue the cat on the roof.
In the end they succeeded in rescuing the cat on the roof.
_{--Longman English Grammar, L. G. Alexander}

However, I think this is a little too simplified, and it could cause confusion, even for advanced learners. In my opinion, it is easier to think that

• could suggests the possibility to do the action, but they probably did not do it, while
• was/were able to suggests that they could do it, and they did it successfully.

And because of that, was/were able to is preferred when we are talking about the successful completion of a specific attempt. Though I believe that this might not be a hard-and-fast rule. The was/were able to always suggests that it's very likely that the attempt was successfully made. The managed to and succeeded in also suggests so, and the achievement is even more definite.

Let's consider the examples:

The fire spread through the building very quickly, but everyone could escape.
The fire spread through the building very quickly, but everyone was able to escape.
(Both versions suggest that they had a chance to escape, but only the second suggests that they really made it, safely.)

They didn't want to come with us at first, but finally we could persuade them.
They didn't want to come with us at first, but finally we were able to persuade them.
(Both versions suggest that we had a possibility to persuade them, but only the second suggests that we really persuaded them, successfully.)

Now the examples from COCA,

I told her to get in line. Did it make you feel better? It did. I finally could say something.
(This means that "I" finally had a chance to say something. Did "I" say something? Maybe, maybe not. Though it sounds likely that "I" did say something. It might not be so.)

Then, when it ended and I finally could get my family back, it came at a price, like suddenly being blind.
(What the text really states is that "I" finally had a chance to get "my" family back. Though it sounds likely that "I" did really get the family back. It might not be so.)

This should answer your question "Are these examples grammatically incorrect?".
Of course not. They are grammatically correct.

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."