Do not import your old wallet private key but to generate new seed phrase on your new cold wallet.

You can add pass jrase which makes your coins to be more secure and safe. But do not lose the passphrase either, just as the seed phrase too. Both should be kept safe and securely in different locations. Also having two or three backups.

If you lose your seed phrase, you coin would be lost. If you lose your passphrase, your wallet would be gone.