After replacing a bunch of these parts, I noticed a bit of white smoke from the exhaust AFTER the car had warmed up.
So next up I replaced the Oil Filter Housing Gasket.
I am guessing that a slow leak in the gasket was causing the coolant and oil to mix. I went ahead and flushed both fluids and the leak doesn't look as bad, but it does explain why the coolant was brown with the first flush.