Back in 2014 I installed Armory and created a wallet, and made a paper backup (and promptly misplaced that paper). Since then I've upgraded and rebuilt my computer twice.
I'm assuming that when you talk about "promptly misplaced the paper" means that you basically hid them in a very specific place only known to you right? I'll assume that you didn't just "misplaced" a word or some of them (although I believe if you've just misplaced a word or two it would be easier to guess the combinations).
Do I need to download Bitcoin Core? Have I lost my wallet? Am I missing a step (or more than one?)
Thanks
You do, because Armory acts just a entity that reads the blockchain provided by Bitcoin Core. As such you'll need to have this blockchain downloaded before you even point Armory to the place where you have it stored. Also, be advised that if you choose to put your blockchain in a different path than the one chosen by Bitcoin Core, you'll need to update that path in Armory due to the fact that, by default, Armory assumes that you haven't changed that path and it won't recognize a new one.
If you shared with us whatever error log you must have it will probably contain something along this line (which would give you a hint of what could be wrong):
BadPath: Invalid blockdata path
As always, make sure you've downloaded the latest version of Armory and please do compare the checksum before installing it to avoid losing your BTC to some scammer.