You do not own any bitcoins if you do not own teh private keys to them.
Newbie here. So if I have a blockchain wallet and have the backup paper wallet, do I go ahead and delete my blockchain account until my next transaction...for now?
My recommendation: get a different, personal wallet that gives you more control, create a backup, test the new wallet with a small amount, and then move all of your coins there.
Now, for the sake of newbs thinking that using Armory, running USB sticks back and forth, etc. is a lot of trouble for a few bits, it might help if if you differentiate between hot and cold wallets.
A hot wallet is one you do your spending from, keep up-to-date, and have ready-to-use. It could be the Bitcoin-Qt client (or Multibit, Electrum, etc.,) or an Android app like Mycelium. (Using an online wallet for your hot wallet is a REALLY BAD IDEA.) You should only ever keep a small portion of your bitcoins in your hot wallet.
A cold wallet is one that rarely (ideally, never) touches the internet, and stores the bulk of your bitcoins. You use it to transfer small amounts to your hot wallet as needed, to receive funds from your hot wallet when its value starts getting too high, and to store your savings. It may be something like Armory, or an instance of Bitcoin-Qt running on a clean, spare Linux system that is normally kept turned off, or even a paper wallet with several keys (created offline, of course.) Different cold wallets have different levels of security; choose accordingly.
If you have a hot wallet and one or more cold wallets set up, you'll have the convenience of ready access to spending funds, plus you'll have extra security for the bulk of your bitcoins. Granted, if you only have a small amount of bitcoin (and plan to use them regularly) it's probably not worth the hassle to create and use a cold wallet, but 3 bitcoins (~$1800) is probably beyond the point at which one should have done so.
And since it apparently can't be said often enough:
DO NOT USE AN ONLINE WALLET TO STORE BITCOINS. If you have an account on an exchange, only keep enough bitcoins there to make your next sale, and make your sale ASAP. Keep everything else in your personal hot and cold wallets; there is
NO NEED to have anyone else "securely" store your bitcoins for you.