To follow her narrative in the novel (to me, that I sounds like a young lady), you can think like this:

I should say something.
= She thought that it would be appropriate to say something.

That is, this thought came to her mind before the saying that she thought she should do.

Compare it with:

I should have said something.
= She thought that it would have been appropriate if she had said something.

That is, at the time she thought it, saying something would be the act that she should already have done. In short, the thought came to her mind after the moment that she should have said it.

To follow her narrative in the novel (to me, that I sounds like a young lady), you can think like this:

I should say something.
= She thought that it would be appropriate to say something.

That is, this thought came to her mind before the moment she thought she should say something.

Compare it with:

I should have said something.
= She thought that it would have been appropriate if she had said something.

That is, at the time she thought it, saying something would be the act that she should already have done. In other words, that thought would come to her mind after the moment that she should have said it.