You haven't lost your bitcoins because they're still in your paper backup. Armory needs to connect to Bitcoin Core to handle transaction data and "cache" them inside Armory so it doesn't have to make so many round trips to Core, and it does so through the ArmoryDB process.

You should download Bitcoin Core (do not launch it) and then when Armory starts it will launch Bitcoin Core automatically and wait for it to finish downloading all the blocks.