OK, I am subdued and educated.
I just loaded my .docm document, did a Save As to make it a .dotm template, and now it stores both macros and quickparts. I'm happy.
I misstated when I said "Although .dotm files are not templates, they can be (and I have) attached them as a template to my main document."
I meant to say
.docm files can be attached to another document. Here's a snapshot of what used to show when I went into Options, Add-Ins, Manage, Templates, Go:
I was able to attach the .docm file after choosing "Attach" by selecting "All Word Files" instead of "All Word Templates." Thanks to your education, I realize I didn't need to do that, but use an equivalent .dotm file instead. But as you see above, the .docm is quasi-considered a template. In this mode, the macros work fine, but the quickparts do not save.
As regarding security, I have the document and template (.dotm) both in the Dropbox folder and they work fine. I am asked when loading the document to enable macros, and then it works fine. Should my security be tighter, and if so, how do I manage that?
In the meantime, BOTTOM LINE, you solved my problem. Many Thanks!!!