What to do? Replace?

Yes, plan on replacing it right away, or at least shut off the power at the breaker for this circuit until you can get a new receptacle.

Look for burnt connections and insulation on the wires connected to the outlet.

I believe you will find four wires, 2 black and 2 white, along with a green or bare wire, and that the connections are loose, and the wiring is feeding other outlets that are being used. The current flowing through the loose connections at this outlet is creating heat that is damaging the outlet and/or wiring inside the box, and that is the odor you smell.

I had a similar situation in my 6 year old home. Thank goodness I could recognize the smell, and pinpoint the problem.

The EC :( that wired the house twisted the wirenut too tight on the white conductors. The wirenut was melted.

Washington,

Just for my own edification :D, how was the wire nut on too tight, and how did that cause a problem? I always twist my wires with a linemans pliers before installing the nut, and I can only apply leverage up to a certain point when installing the nut by hand, hmmmmmm... never heard of this problem before

Whenever a wire nut melts and wires are burned, it's always a case of a loose connection, not too tight of one. And Shader, every good electrician that I know does exactly what you do, mechanically tighten the connection with linemans pliers before installing the nut. It's the proper thing to do.

Yeah, I only told half the story. The wirenut had too many conductors twisted waaaaay too tight ---breaking through the wirenut. -The break-through caused arcing---which melted the wire nut.