Well, while I was typing all of the below, I kept responding to prompts and I eventually got a browser screen that would stay open, and after 2 two-stage verifications, it told me I needed to create "application-specific passwords". I thought "oh great". But all that consisted of was naming my device (I chose Google Nexus...) and push the button "Done". It generates a 16-character password that you don't even need to write down (I did and didn't need it), and the next thing I know, I am logging onto my library's website, browsing IMDb, the whole nine yards!
I've no clue what I did wrong in the first place, or what I did to fix it. I'll miss it when it poops out completely!
Thank you for your answer! I hope I provided some good information for someone else in the future.
I have a triangle in the upper right corner that when pulled down says "Account Action Required".
I push that, and I get "Couldn't sign in
You typed an incorrect password or your account has changed. Re-type your password. If you forgot your password and need to recover your account click here."
When I push "here", it tries to open a browser to Google saying "Having trouble signing in?" with 3 options to select from, but that quickly goes away and returns me to the "Couldn't sign in" screen. Which has, BTW two selections at the bottom, Try again and Cancel.
If I push "Try again", I get a key pad, and I enter my current Google password, paying very strict attention to case.
I just tried logging in again, and it took me to a Google login screen, and I entered my current password, and it sent me a verification # to my phone in a text message. I entered the code, and the browser window went away and I was met with "Unfortunately Exchange Services Has Stopped" With two options: "Report" and "OK". Or if you are not