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There is just the gas fee for moving each crypto. Doesn’t matter what wallet you’re sending to (Trezor, another Exodus, or an exchange), you’ll need to pay the gas fee. If you’re unfamiliar with this concept, it’s because your exchange of choice is simply hiding the fact that they’re paying that same gas fee from you, by adding it as a different fee.
You can get an estimate on Blockchain explorers, here is Ethereum’s, it will also apply to any ERC-20 tokens (UNI, AAVE, USDC, etc). https://etherscan.io/gastracker See the „USDT: Transfer“ for the estimate to just move a token.
Doing it at night (USA) or weekends is cheapest, but generally things are cheap most of the time currently due to bear market.
To answer your statement directly it’s less about registering the new address, and more about just the fact that you’re paying the miners to acknowledge the fact your coins are moving from one place to another.
However, some coins do require registering an address in advance, or at least holding a minimum balance such as Stellar. It’s likely that most of the coins you’re talking about will be either bitcoin, ethereum, or an ERC-20 token on ethereum, which don’t have that complexity.