Question on mixer dApps

Howdy folks,

I have a question regarding the mixer dApps (e.g., Tornado Cash, Onion Mixer, etc). I know how mixers work, but I'm not 100% sure I know how to use them properly. Consider the following scenario:

1. I have an address A that can be traced back to me, and I want to send my funds from there to a new address that won't be traceable (address B)
2. I go to a mixer dApp, deposit the funds from address A and get a corresponding note
3. Now, it's time to withdraw the funds to the address B. It can be done by any address calling a specific function in the smart contract. However, to do so, the address that I'm using should have some amount of ETH to pay the transaction fee.

Here is the dilemma — ideally I'd want to use address B to initiate the withdrawal and call the contract, but address B is brand new and does not have any ETH. Hence, it cannot pay the transaction fee and cannot be used to call the contract. If I use address A that has some residual amount of ETH, then it will defeat the very purpose of the mixer since address A will be linked to address B in the transaction details.

Any idea on how to solve this riddle?