I found the problem. I turned out to be AntiVirus Software related -
which is why it worked on some machines but not others. The AV
software we're using is Kaspersky (which is also what the other user
was using in the thread linked to in my original post). I'd tested out
the AV by disabling it, but that didn't fix it. However, for some
reason, when I change the Real-Time settings for Kaspersky to
High-Speed, the Macro should work correctly. easy, and so annoying
fix. After the setting change, I was also able to keep kaspersky
enabled while running the excel macro. Open for Append works fine now
in creating and deleting the files.

-Michael