Why won't Notes Sync with Gmail?

Why won't my iPhone's (and corresponding ones on the computer) Notes app let me sign into and sync with my Gmail account(s)? It is telling me the user name and passwords are incorrect?

I have tried reading all support topics and checked with 3 Apple Care representatives and had no luck. I need to save my Notes, NOT in the Cloud. I plan to discard my iPhone. I need to backup the Notes into Gmail. I will be getting rid of iPhone. I cannot do that until I save my information, again NOT using Cloud. Cloud has caused me all kinds of problems. Why won't it just sync and save to Gmail?