So, if you have some coins in a wallet that you received into an address and then if you spend some of those coins, the remaining coins should be moved into a new address that has no previous coins?
No. You should move your Bitcoins to a new address that has no spent transactions. It's fine if you move it to X address that has thousands of incoming transactions without any unspent transaction because the key is not public yet. It will also increase your privacy as long as you can mask your transaction with Coinjoin or mixer like Chipmixer. CMIIW.
Read more here:
https://en.bitcoin.it/wiki/Quantum_computing_and_Bitcoin