OP, I'd sure be looking very closely at any signs of overheating,
checking for loose connections, signs of oxidation/burn insulation,
etc., etc.

If I didn't find something easily correctible I'd recommend a pro,
espectially if this is new symptom. Could be early warning sign of
potentially serious trouble.

Well, I opened every outlet, junction and switch box on the entire
circuit to check the connections. All were pretty secure except one,
which I re-tightened with a new wire nut. All the connections looked
like copper to me.

I unplugged a "permanent" extension cord the previous owner had drawn
through a basement window and left outside.

The lights still flicker. I am waiting on the power company to check
the house connection.

Tonight I'm thinking of turning off the main and having a look inside
the breaker box.

I don't think it's grease - it's very oily and clear.