Is it possible to "send" Bitcoin to yourself (new wallet) without announcing it to the network and without paying any fee?
The network comes to consensus on how much Bitcoin is held at individual addresses. If you want the network and the blockchain to recognize a certain amount of Bitcoin is held at a new address rather than an old one you can't avoid broadcasting a transaction and paying a fee. Of course there are offchain methods of transferring value like Lightning and Liquid between addresses but the Bitcoin blockchain does not know of these transfers.