Thanks everyone for the suggestions. I pulled out one of the plugs, and it's one of those self-contained types, which does not use any screws to hold wires. Instead they are in these very tight metal crimp type connections. It's basically this plug:
I didn't notice any melting in either outlet. Do these types of outlets go bad, even if visually they look fine? Or does a bad outlet usually show some sign of melting/burning? Does anyone have suggestions for removing the wires on these types of outlets? They are pretty secure. I'm thinking I can unplug each outlet and try turning on breaker to see if it's a particular outlet.