Peter, thank you for your input. According to the documentation I have read, the "NO_SMS_ON_DRIVE.SMS" file only does its "magic" before the distribution point is established or before a drive receives any packages on it. The issues I experienced were experienced before I created that file.
What I ended up doing is removing the reference to that DP from all packages, then removing and reinstalling the Distribution Point role. As the drive "D:" was no longer the official drive that the distribution point was using, it did not remove the package or signature files from the original drive "D:", only the ones on "F:". I did have to manually remove the SMSSIG$ share as it was still pointing to "F:" (which involved me having to re-create the directory as the share was "orphaned").
Once I started reassigning packages to that DP, the files that were already on the "D:" drive seemed to do what a package "refresh" does- which seems to be a shorter process (maybe I'm wrong). The packages that were on drive "F:" had to be re-copied to the DP through the distribution manager process.
It all happened overnight, so I came in this morning to a happy and fully functioning Distribution Point. And now that I have the magic "NO_SMS_ON_DRIVE.SMS" file on drive "F:", SCCM is behaving itself and only using drive "D:" for the packages.