i get your point, what you are trying to say is that if you have two bitcoin address and one is known by many, and you received bitcoin to wallet A and you want send your coin from wallet A to wallet B without without trace on the blockchain that will it be possible right ? well there is no transaction made in the blockchain technology that is not store in the blockchain or public ledger. the only way you transaction can remain unidentified is by using mixer service. although i have not made any transaction using mixer but from discovery mixer is the only way to make a transaction untraceable.
Even with a mixer, your transaction can only not be traced back to where it came from, but the coins will be visible for public view on the blockchain. The OP question is just kind of complicated if you ask me, as if he is asking two different questions which require the use of coin control if he wants to spend bitcoin from one wallet which shares the same key with the existing one without spending from the other one and also emptying the old wallet without where the coin went to being traced. Both actions need to be carried out, and privacy maintained like the OP wants. He will need privacy tools to get that done, but they will be visible on the blockchain.