The fire spread through the building very quickly, but everyone was able to escape.
They didn't want to come with us at first, but finally we were able to persuade them.
I told her to get in line. Did it make you feel better? It did. I finally could say something.
Then, when it ended and I finally could get my family back, it came at a price, like suddenly being blind.
All four of the examples you have there are grammatical, and they are all acceptable in their specific contexts.
CAVEAT: This post is written on the fly, off the top of my head, etc., etc., and so, be forewarned.
There can often be a difference in meaning between the two expressions: "BE able to Verb" and "could". The context that they are in will often be a significant factor.
First, let's look at the four directly involved clauses in isolation:
Everyone was able to escape.
We were able to persuade them.
I (finally) could say something.
I (finally) could get my family back.
1/2: "BE able to Verb": The first two (#1, #2) could strongly imply that everyone did escape (#1), and we did persuade them (#2). If that type of strong implication is not immediately cancelled, or if the context had not already cancelled it, then that implication is then assumed to be a fact by the reader.
Though, the strength of that implicature could depend on the "Verb". For instance, for #2, it would be relatively hard to create a context where that implicature is not realized, that though we had the capability to persuade him, we chose not to. But for #1, that is not the case, as it would be easier in this case to come up with a context where that implicature is not fulfilled, e.g.:
- I cut their ropes, thus freeing all the captives. Everyone was able to escape, but they just remained there in their seats. They looked like plastic mannequins. "Go, leave," I said. They gave me blank stares.
Creating a corresponding context for #2 would be a bit trickier (if possible at all):
- . . . We were able to persuade them, but we chose not too. . . . [???]
3/4: "could": As for your last two (#3, #4), it seems that there isn't any implicature (or else there is only a weak one) that "I" actually did say something (#3), or that "I" did get my family back. All those two are saying is that the possibility was there: for me to say something, for me to get my family back. There isn't anything there to strongly imply whether or not I did. The context they are in will probably have to explicitly say if I actually did.
Now, let's look at those four versions in their original contexts:
1) The fire spread through the building very quickly, but everyone was able to escape.
The context makes the very, very strong implicature that everyone did escape. If everyone didn't escape, then the writer has to immediately cancel that implicature.
2) They didn't want to come with us at first, but finally we were able to persuade them.
The context makes the implicature too strong to be cancelled. That is, it is a fact that we did persuade them to come with us. But, it is still an implicature (a cancelable one) that they did end up coming with us--for it is possible that something then happened to prevent that.
3) I told her to get in line. Did it make you feel better? It did. I finally could say something.
It is the context--that I had earlier told her to get in line--that makes it a fact that not only could I say something but that I actually had said something. But a different context for that last sentence can easily show a situation where I didn't say something:
- There was a gap in the conversation. All of her friends were temporarily occupied with paying the waitress. I finally could say something. But now, I thought better of it, and I wisely decided it best to keep my trap shut.
4) Then, when it ended and I finally could get my family back, it came at a price, like suddenly being blind.
Here, the context has it pretty much open-ended: that is, the reader doesn't yet know if I actually did get my family back. Here's a continuation where I don't:
- Then, when it ended and I finally could get my family back, it came at a price, like suddenly being blind. But I like seeing, like watching TV and going to the movies, and reading books, and driving my car. And anyway, my family never much noticed my absence before, and so, I turned down the devil's not so generous offer. Living alone ain't so bad, and it's got its advantages. Like full control of the TV remote.
CAVEAT: I did this post off the top of my head. And now, maybe I'll go and do a little checking to see if I made any major goofs. . . .