Hey Frustratedmom77,
Thanks for using Apple Support Communities! I certainly depend on my Notes to stay organized, so understand how important those can be! I am happy to provide some information about Notes, and look into options to recover those.
You stated that you attempted to restore your Notes from a backup. When restoring your device, the current information on the device is overridden by the information in the backup. Any Notes that were not included in the backup would be removed. Are the Notes that you were trying to restore on your device?
Also, do you use iCloud for Notes? If so you may find your Notes by logging into iCloud from a computer. You can find more information here: iCloud: What is iCloud Notes
If you're not using iCloud for Notes, and still have access to your other device, you can choose to enable that now. If you're using the same Apple ID on your new iPhone, you'll be able to see your Notes on that device as well.
Thanks.