You can do a very similar procedure with electrum as well.
When you do file>save copy in Electrum, the copy will be encrypted if the original was. Of course any wallet you put in the cloud should be encrypted, a few years ago dropbox had a security issue that allowed anyone to log in to anyone else account without a password. The issue remained for a few hours.
Only the hand-written seed should be unencrypted, I would not recommend encrypting it as if you forget your password you'll have no way of accessing your funds, you should always have the means to access your wallet in the event you've forgotten your password.
This would be essentially the same thing you would do with armory, except that armory is much more encouraging for you to back it up this way.
The problem is though, if you happen to get diagnosed with amnesia, you won't be able to access your Bitcoins to pay for treatment as you'll have forgotten all your passwords, so you should always have a way in to your wallet without a password in case you forget your passwords, which is why I recommend an unencrypted handwritten seed. If you absolutely must encrypt the seed, then you should at least store a password hint with it and you shouldn't use a really high iteration count so if you forget a character or two you'll be able to bruteforce your way in. Obviously such a seed should be kept in a very safe location if physical theft is an issue.
Armory also tries to force you to make at least one unencrypted backup for this reason. Without a way of getting into your wallet without a password your wallet essentially becomes a brain wallet.