Preview app does not save notes immediately in PDF file after clicking cmd+s

Ok, Please check in safe mode if the issue persists.

How to use safe mode: https://support.apple.com/en-in/HT201262, Which may take more longer than your usual Normal boot

Note: Safe mode prevents your Mac from loading certain software as it starts up, including login items, system extensions not required by macOS, and fonts not installed by macOS. It also does a basic check of your startup disk, similar to using First Aid in Disk Utility. And it deletes some system caches, including font caches and the kernel cache, which are automatically created again as needed.

If you have issue in safe mode as well please contact Apple Support. They may provide you more specific step to resolve this issue or may even ask you to re-install macOS.

Ok, Now we need to check if you have the same issue with another user.

Please follow this link Test with another user account.

Note: If another user account is already set up on your Mac, you can simply log out of your account (Apple menu > Log Out) and log in with the other account. But the best way to test is with a newly created account.

Please check you have read & write permissions on the pdf files. Also check if the issue is persistent with other users as well.

If it is not related to having right permissions, suggest you to check with other app(adobe acrobat / any which supports pdf modification) that you are able perform your edits.

If you are able to modify/edit notes & highlight with other apps then the issue is with preview app handling pdf files.

Are the pdf files encrypted / password protected? If they are then you can't modify the pdf.