After suggestions that I try drybrushing instead of painting on top of airbrushed layers/basecoats, I decided to give that a shot.
I started with a coat of Vallejo flat brown:
Then some of Vallejo Air Fire Red:
Then washed with a mix of GW's Druchii Violet and Reikland Fleshshade:
Then drybrushed with Khorne Red:
Then drybrushed with a bit of Mephiston Red:
Then drybrushed with a tiny bit of Vallejo's Vermillion:
You can see here that I attempted to paint some highlights with vermillion, but it didn't really end up looking good. Maybe I'm not thinning my paint enough (though I'm definitely thinning it, maybe too much?) but when I paint on top of airbrushed layers it just doesn't look good. Drybrushing seems to be the key on top of anything painted with an airbrush.
Then I experimented a bit more using nihilakh oxide.
Vallejo bronze sprayed on black primer, nihilakh oxide painted into recesses, nuln oil wash, and then drybrushed warplock bronze to test the color, then highlighted with vallejo bronze again.
I think if I start with warplock bronze as the basecoat, then use even less nihilakh oxide, then nuln oil, then use vallejo bronze as the post-wash drybrush/highlight, it will look just the way I want it to.