But for the sake of your anonymity then just have to choose another address with that private keys that you own.
Using fresh addresses is good for privacy, but just that alone does not make you anonymous. Even the BTC blockchain is public and transparent and then if you send your funds into centralized exchanges, you obviously no longer have any privacy to protect. If you want anonymity or a much better privacy, you must first be running your own node, once you connect to third party servers, the host already has information about your IP and your BTC address, you must also use p2p exchanges and privacy solutions like CoinJoin implementations.