right now the simplest way to get your bitcoin cash is indeed to import or sweep your core private key into a new wallet.
i did it from a paper wallet to coinomi because i didn't want to screw around with unknown wallets or the bcash blockchain. it deposited and sent fine.
send your bitcoins back to your wallet only when you've created a new one.
Thanks for the info. Because I do not have a smartphone, I will have to use Bluestacks unless you know of any Bitcoin Cash-recognizing desktop wallets (please let me know if you do). However, Bluestacks only works properly when my computer runs on Windows 10 (it is currently running Windows 7). How can I save the private keys for my Bitcoin Core wallet so that I may upgrade?