Samsung notes has an option in its settings menu titled Sync With Samsung Cloud.
This is the default method we use to automatically sync /backup our notes. It has other options for saving and sharing our notes using several different file formats with optional apps but it doesn't sound like you've used any of these to save your notes.
In this case I think that the only option seems to be making sure Sync with Samsung Cloud is turned on and make sure that it's syncing properly by opening Samsung Cloud, Other Synced Data and scroll down to the Notes App. Check the date that it was last synced. There's also an option at the bottom for manually syncing so you can watch it sync in real time.
