Oh hey! There's an app you can pay money for that just notifies you of incoming AOL mail. I am apparently not the only person with this problem.

The notifier for the mail app appears to be working better now - it's not quick, but it's not deadly slow. And with this app, I think I'm covered.

Now I just need to get work email working. It won't set up on its own - I need to either get my work IT department to do it, or get someone else at work who's done it successfully to set it up. (My understanding is that our IT department goes back and forth about whether it's allowed to use your own device for email, so on any given day they may refuse to help with set up.)