What happens to your old wallet when you reconstitute your private keys on a new cold wallet? I'd like to move over to a bitcoin only wallet. I'd add a passphrase to the new wallet for an additional layer of protection.
Nothing would happen to the old wallet since you would be using the same private key, but you shouldn't do that. Instead, sweep the private keys instead of importing them. You wouldn't want to risk importing a private key/seed that might have been compromised to a cold wallet. Which cold wallet are you going to use?