Once your cold wallet gets access to an internet connection it is no longer cold. You would need to make a new offline wallet address (the same way you made the original cold wallet) and send whatever fraction of the $10 that you didn't use to that new address.
True.
But then OP could sign the transaction on the offline machine, and broadcast the transaction on the online machine.
In that case, he won't need to load the private key on an online machine.