What about a third option, just send it from one wallet to the other like you were sending coin to someone?
This is exactly what "Sweeping" is.
Sweeping" is only a term that refers to the act of sending every coin available in X address to another. Sweeping or manually sending all your coins, it's all the same.
So, most guides will suggest you to:
a. Import your private key in another software and keep using your old address.
b. Sweep (send) all your coins to another recently created address.