After tweeting about my flat feet, I got many helpful responses, one of which has led me to Newton Running and the shoes below:

I plan on heading over to the Newton shop in downtown Boulder in a few hours, where I'll try them on and see what the scoop is. I understand that Newton espouses a different style of running than "normal" running. Something about striking the ground with your forefoot instead of your heel. 

I don't know much about all this, but if I can pull off wearing bright yellow shoes, it will be gamechanging for people who say things like "gamechanging."