I seem to have resolved the problem by saving the template as a .doc and then reconverting it to a .dotx. As previously mentioned, the template has no modern bells and whistles, so, while I did get a message about compatibility when saving to .doc, only a couple of simple text boxes must triggered it and they survived visually and functionally for our purposes.

Prior to this, opening the template or a document based on it in repair mode worked for my 2007 computer, but not for two other users'. Thus, I took the above-cited step, hoping it would save me, as it seems to have, from having to recreate the entire template in Word 2007.