Transferring many bitcoins without restoring a seed
You would want to use a mixing service. Essentially, you would send your 10 coins from your old wallet to the mixing service and get the same amount out to the new wallet. This effectively anonymizes your coins if the mixing service is doing its job correctly.

A similar effect can be done by sending your coins to an exchange from your old wallet and taking it out to the new wallet. Obviously, the exchange would be able to connect the two addresses, so it wouldn't be as anonymous.