All you need to do is store the public key. This key is good for giving to anyone. It can only be used to query the balance in the address and/or send BTC to you. Basically you'll only need to connect the wallet to the a machine with an internet connection once you need to move the coins. But remember that once your private key hits the net then your storage is no longer "Cold" and you should forget about any benefits associated with offline storage.
If you use the private key in a wallet connected online, you should move the rest of the coins to a new address generated offline if you'd like to maintain the benefits of offline storage.
Just for my understanding...
If you follow the procedure here
https://electrum.readthedocs.io/en/latest/coldstorage.html for signing transactions in the off line wallet and then return the signed transaction to the online wallet,
does this expose my private key?
This is a good way to create transactions without exposing a private key, but if the "Offline" machine connects to the internet at any point (even let's say after uninstalling the wallet and deleting private keys from memory) then the address no longer cold storage.