Figured it out! Whew! Only took hours of searching and experimenting.

It seems there is a bug in Word 2016/Office 365 (may affect other versions): if you have both bookmarks AND you restrict editing except for certain text areas as described above, then you will see the vertical lines I describe. Remove the bookmarks and the lines go away. Fortunately, I don't need those bookmarks. Others may not be so lucky, and will have to use form fields and lock the document for forms. (Which is what I would have done from the start except I inherited these docs from someone else. I'll redo them at some point in the future when time permits.)