So I typed in my sixteen words and the coins appeared in this new wallet.
You probably had 12 words, not 16 words.
I now have the novelty that I can see them on both the old PC in the old default wallet and the new PC in my named wallet.
Note that with having the same wallet in two different devices, you increase the risk of getting hacked.
I plan to eventually move my coins to a Trezor wallet. But is there any way I can disable the old default wallet in my old PC, before I get rid of the old PC entirely?
You can delete the wallet file, but note that even if you do that, you won't remove the wallet completely from your hard drive. There are file recovery tools that can recover the delete files.
The best thing you can do is to create a new wallet in your new device and move all the fund to that.