I can't delete my quotation marks or apostrophes

I'm not even sure that's how you describe this problem! I put a sentence in quotation marks: "like so" or use the apostrophe like so: 's, and then sometimes when I go back and delete the quotation mark, the computer will start to delete the text instead. Not only that, if I try to add more to the quoted text, it will add the words outside of the quoted text. Like so: something I've added before the quotation will be here"Quoted text". The only way I can fix this is to highlight the text before and after the quote and delete the highlighted text and rewrite the sentence.

I don't know how to fix this and I'm not even sure if anyone will understand what I'm talking about without actually seeing what it does. I gave my most recent document to my editor and it started doing it to her in my document. She's never seen anything like it before. It's happened in the past, but it's becoming more frequent. Help?