Thank you for these answers. So I'm encouraged that basically you're saying it's doable, at least in concept. I don't mind a few extra steps.
Some clarifications:
Actually, sweeping is recommended since when you export the private keys, you are at more risk of them being stolen. Sweeping makes sure that even if someone steals your private keys, there won't be any Bitcoin for them to steal. It won't work if you are offline.
I don't see there could be more risk of them being stolen.
I'm planning on exporting the keys from Qt and importing the keys to Armory as follows:
(1) Copy the Qt wallet.dat into the offline machine, fire up Qt, and use the command-line interface in Qt to export the private keys.
(2) Import the private keys to Armory - also on this same offline machine.
So... How could anything get stolen in this scenario?
It is the fact that you have decrypted private keys available. In reality it is really unlikely that they will be stolen, but you should still be careful when handling exposed private keys. An extreme example (not likely to happen at all):
You have a virus on that offline machine. The virus recognizes when you have a Bitcoin private key in the clipboard and that you happen to have a USB flashdrive plugged in. It copies itself onto your flashdrive such that it automatically runs and installs itself on your online machine. It also copies over a file that contains your private key. When you remove that flashdrive after you are done and go to the online machine to broadcast, the virus installs itself on your online computer and sends to the hacker the file with the private key. Someone has just stolen your private key.
The problem is that bitcoin core uses compressed keys now and armory only supports uncompressed keys, so the keys from Bitcoin core won't be able to be swept or imported.
This is actually a really old wallet - I think it's about 3 years old. So I think it doesn't use compressed keys.
Possible alternative:
I guess I could also simply sign a raw transaction from Qt itself and then broadcast it to my new Armory offline "permanent" wallet, from a site like blockchain.info/rawtx. This would obviate the initial step of importing to Armory, into a "temp" wallet, as mentioned in my step (2)(a).
However, since I'm going to have to learn anyway how to offline-sign and broadcast transactions in the future using Armory, and since I haven't learned how to do that yet using Qt, I figured I might as well just get the private keys into Armory, and then have an all-Armory all-offline solution for getting the funds from Qt to Armory, without having to learn how to offline-sign raw transactions from Qt.
I would only do this alternative (offline-sign a raw txn from Qt, and broadcast from blockchain.info/rawtx) if the keys from Qt actually do turn out to be compressed. But this wallet is really old, so I think they're non-compressed.
You will find out when you export. If you export the private keys and they start with a '5' then you are fine. Those would be uncompressed keys. However if they start with 'K' or 'L', then you can't use those and since those are compressed keys. In that case, you would need to offline sign with Bitcoin Core.