Learn English – Can “can” be used to imply “become being able to”

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.

Both sentences are grammatically correct. The second sentence uses a backshift to indicate that we are talking about a hypothetical situation- this is used for all verbs except be. The first one uses a proper subjunctive were (the only one!).

The first (with were) is, in my opinion, stronger in suggesting a hypothetical situation. In this situation- Sandra's mother is probably no spring chicken and already 'ailing'- we are probably talking about something that could really happen, so the less strongly hypothetical version became is more appropriate.

The were version is slightly more formal than the became version, but both are a lot more formal, and less widely used, than one using got if you make the second part less formal too, you get:

If Sandra's mother [got] seriously ill, Sandra [would never] forgive here husband.

This NGram graph shows the trend toward the less formal (got) usage: remember that Google Ngrams is based mainly on written English, and for informal usage it lags a long way behind spoken English.