Program unable to access \Appdata\Local\Temp (Permissions issue?)

You said it works on the new administrator account, but then you also said it doesn't work on the new account.  So did it only work once?  Or are these two separate accounts we're discussing?

Secondly, can I ask if you are able to open your Temp folder yourself (not using the Finale program)?

That's very odd that the situation would change day to day.  There seems to indicate that there is a change in the environment that is conditional. Maybe another program running in the background that was only running one of those days. Or maybe there's a helper service that should have been running but wasn't.

In any case, if it is working now I suppose I won't recommend any actions that would break the situation.

However, I want to point out that many times with software that was originally built to "expect" administrative privileges, it would have problems when running on newer operating systems like Vista/7 onward (Windows) or Mac OS X onward (Apple).  New operating systems block programs from doing "admin" things, even when you the user are logged in as admin. This is because anything running while you're logged in can do anything you can do. Thus, programs running while people were logged on as administrators is what causes the ransomware attacks we see today. Since that person has access to the files, any malware has access also.  So in modern operating systems, even administrators can't readily modify system file locations or shared places like temp files.

Two things you can try:

1. While logged in with the "problem" user account [Paul], try to right-click and Run as Administrator for the program.

and/or 2. Uninstall the program completely and reboot.  Then, go to the installer setup.exe file and right-click that and run-as-admin.  This is different because the installer routines itself will now be able to properly perform all the installer commands that may not complete successfully without the 'assumed access' to perform the admin tasks.

If both still fail, it can be beneficial to run the installer setup file in "compatibility mode" which you can do from the same right-click menu on the setup icon.

Right now I have two accounts: my primary account which I've used for years [Paul], and the new one I created a few days ago [Paul2]. Both are Administrator accounts.

They asked me to create a new Administrator account and try running the program on that one, I named it [Paul2]. The first time I ran it on [Paul2], none of the errors I listed occured and it seemed to function as it was intended to - so I reported that back to their support team. The next day I logged on as [Paul2] and opened the program but it behaved just as poorly as it does on my original account [Paul] (slow, not responding, all the errors). However, right now as I write this Finale is working perfectly on [Paul2]. 

I am able to open both Temp folders on [Paul] and [Paul2].