How do you paint shadows without the piece looking "Dirty"?
Don't use black. I use Payne's grey to darken colours. The correct way to darken a colour is to use it's complimentary colour but that does not always work, so I use Payne's grey. What you also can do is if the paint is transparent, paint the dark areas first and then the main colour over it. That is what I've done on the shell of this ladybug.

I'm with AndreZA, darken your colors. (Don't just spray black on top.)

I was always told: There is no black in the natural world, just darker values of colors. Highlights are the same way, unless it's wet or has a hard's never white. I have broken those rules, sometimes I get lucky...sometimes I pay for it, lol.(In all honesty, colors are still a big pain in the neck for me)

I do my shading with a darker version of my base color. In this case I darkened most of this with transparent red mixed with my base color, I added just a little blue to my shadow color for the darkest areas.
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Watch Airbrush Tutors video on flesh tones, he talks about making different values of the same color.