Thanks, Meanderer, that vid seems to be for a newer version of Word than 2003. But after googling (which just told me to do what I already tried with no luck: selecting the text and making sure Black and no Strikethrough is selected) and after stumbling around all over the place in the document, I think I finally found it: in Word 2003, in the document you go to Tools, Options, Track Changes, and then make sure that "Insertions", "Deletions", "Formatting", and "Changed Lines" are all set to "None".
So I did that and everything's okay now. But what's so darn weird is that this is an old document I've had on my computer for a long time, this has never happened before and all of a sudden today it just started doing that, and it's hard to believe I would've accidentally made all the changes above to track changes in Red & with Strikethrough. IDK, maybe there was some kind of internal Word update but you wouldn't think so with that old a version of it. Oh, well, onward and upward.