You DONT need to send your BTC out to a different wallet. All you need is a new address inside your existing wallet. That address has a different privatekey, and that is enough for forks that dont employ replay protection (all the ones we support do).
be careful with your funds.
If you recover through this method using the Recovery Phrase (or seed phrase, whatever you call it) you expose all the private keys in that wallet.
If you use only the private address of each public address, you are right. This is not the case.