First, the overriding assumption below is that the quoted statement is not a question or an exclamation, in which case you would use the ? or the ! within the quotes at the end of the sentence. But assuming this is a simple statement that would normally end in a period ...
As a 50yo native U.S. speaker, the main grammar thing I see here is that we always learned "comma-then-quote" if anything more is to follow.
From that point of view, I would consider both of the above examples incorrect grammatically, strictly from a punctuation point of view, but all of the following would be considered correct (including the spaces, which you seem to already know):
- “I told her my secret,” she said.
- She said, “I told her my secret.”
- “I hate her,” said Laura.
- “I hate her!” Laura said.
Where there are commas, they are always before the quote (and if the sentence ends with a quote, the period is present inside/before the quote).
And I would never put a comma between the verb & noun, if you were suggesting:
“I told her my secret.” she, said.
Just in case you had a passing thought about that, that would not be correct.
But as for word order, I would say that "she said" sounds noticeably more natural than "said she." I would not call the "said she" word order "arcane," but it is definitely not "modern." I would expect to read "said she" from Shakespeare or even Tolkien (1930s-40s) rather than a modern Rowling or Stephen King novel.
But I also agree with your "said Laura" (interchangeable with "Laura said"), so it is the pronoun that is the sticking point in that word order, at least in common use today.