You vs. Them (Accounts)
In this answer, I discuss two Google accounts from the perspective of an account belonging to you, versus one belonging to another user. In your question, the "other user" is your old account containing the playlists. This you/them approach makes my explanation simpler. The fact that you control both user accounts is not relevant to the solution.
Whose playlist is it?
When you add another user's playlist to your account, the owner of the playlist doesn't change, you are merely adding (displaying) their playlist in your account. In other words, there remains only one playlist that they are sharing with you. That is why changes to the playlist are being synchronized.
In order to make a completely independent duplicate of another user's playlist, you need to add the playlist's contents to a new (or existing) playlist that you own. What follows are the simplest instructions I know to accomplish what you want.
Pixel Help Playlist EN - by Google Help is used in example URL's and screenshots.
- Navigate to the URL of the other user's playlist.
- Open the playlist in Google Music by replacing
music in the URL
- Click on the three-dot menu then "Add to Playlist"
(Do not click the "Add to library" button)
- In the "Add to playlist" menu click the "+ New Playlist" button
- Click the "Create" button in the "New playlist" dialog, after entering the "Title" and confirming that the "Privacy" settings are correctly set
The playlist is now duplicated and the new one is completely independent from the original, but contains the same content.
Default Sort Order
When adding a video to a playlist, YouTube's default behavior is to add it to the bottom (end). This is also the behavior if your playlist has been sorted manually. If however, you apply one of YouTube's optional listed sort orders, that selection will continue to be applied to the list until you make a different choice or a manual sort is performed.
Set the Sort Order
- Return to your YouTube library located at https://www.youtube.com/feed/library
- Click on the "View full playlist" link of the playlist just added.
- You now have two types of sorts to choose from:
- Listed Sort: Click on the "Sort" menu at the top to select one of the listed options to be applied to the playlist moving forward
- Manual Sort: Set the order manually by dragging videos into position using the grab handles located on their left side. Future additions to the playlist will default to the end (can be reordered).