Installed Bitcoin Core 0.21.0 and Armory 0.96.
Added an external HD. Moved bitcoin core to d:\bitcoin. Left Armory installed on the c: drive. That's why I left the pathing to both drives, depending on whether it was for bitcoin core or for Armory. If it's the best approach, I will uninstall Armory 0.96 and reinstall 0.96.5 to run with the new version of bitcoin core.
Yeah, I suspect some of the issues might be caused by trying to use 0.96 with Bitcoin 0.21... the older versions of Armory don't play nicely with newer versions of Bitcoin Core
Removing 0.96 and installing 0.96.5... get it running (and perhaps consider a "rescan and rebuild databases") and then see how things stand.
Should Armory be installed on the d: drive or the c: drive for the upgrade?
It really should not matter... theoretically Armory can be installed anywhere... and it's data directory can be configured to be anywhere.
However, for simplicity, it's probably easier to simply install Armory using the 0.96.5 installer and leave it's install (C:\Program Files) and datadir (C:\Users\USERNAME\AppData\Roaming\Armory) locations as "default".
After the system is running, it sounds like I could either take the signed transaction and convert it to RAW HEX, then broadcast it to the network or import the wallet from the 2014 system with the BTC and go on with business as normal. Will the second option work? Are there any known issues with wallets or with anything else in between versions that I have been running since 2014?
Depends on how you wish to import the wallet... trying to import the wallet files themselves might cause problems... So, I would say that you'd be better off restoring from your root key (paper) backup.
If the signed transaction you have is still "valid" (you would be advised to triple check ALL the details of it, like output address(es) and amounts etc) and you still wish to broadcast it, then that is just as easy to broadcast from one of the raw transaction "push" services (coinb.in, blockcypher, btc.com etc).
I guess the question is, are you just trying to get your coins out of Armory and then move on, or are you wanting to continue using Armory?