As an aside, I'm shocked that Armory creates paper wallets that are completely dependent on a functioning online Armory running which includes having the entire 30GB (and growing!) blockchain, rather than paper wallets that can be restored by any wallet or standard encryption.
You're confusing the paper backup and bitcoin private keys.
- The paper backup is a seed that allows you to recover every private key ever created and will ever be created by your wallet. It's called a "deterministic wallet," and Armory was the first wallet to implement that which is why it's not compatible with anything else.
- The individual private keys are what hold the actual money. Those are derived from the paper backup. Backing those up only backup the coins they currently hold, but don't help you if you use more addresses.
In your case, you want the private keys. Start Armory in offline mode (there's usually a shortcut installed with it specifically for offline mode). Then load the Armory wallet and go to backup options. "Export Key Lists". There you can get the first 100 raw private keys that can be imported into another app (you might have to click "Include unused" since Armory will think that all the addresses are currently unused).