I'm no painter, but what I have seen others do is rather than trying to paint the white fur, they paint the shadows, fine lines, etc.. over the white background, essentially leaving the white where they want it - kind of like painting it in reverse ... rather than ading the color you want, you are adding the highlights you want.
I know that's how it is also done when painting onto stained glass ... the black paint is put over the glass where you want the shading, lines, etc.. leaving the color of the glass to show through everywhere else. It works very well with glass as the shaded (lighter) sections of black are tinted by the light coming through the glass.
You might want to experiment with other medias ... maybe charcoal, pastels, or something for the fur highlights ... something to let the white bleed through.
The biggest thing I think is not to get too carried away trying for every shade of grey. If you can, I'd say stick with straight black, and experiment with how much of it you use, from a subtle hint (almost/virtually dry brush) to a heavy (wet) brush stroke - it will keep your shading much more uniform then.
Otherwise ... I'd probably be inclined to stick with only the "darker half" of the spectrum and avoid anything that was more than 50% towards the white end of the scale. That would eliminate the noticeable differences in the white of your background and the white of your paint - no two whites are ever the same ... and with mismatched whites, the lighter grey's that tend to get a blueish tint to them will really stand out when you step back.
From what little I have tried (while trying to teach myself airbrushing techniques) with something like what you are doing (applying varying levels of black over a color), I found that less is usually more. Just enough highlights & definition to give it shape ... after that, the viewers eye will fill in the minute details.
