Another way to desaturate a color is by making tones of the color. Toning a hue down is adding a gray that is the same tonal value (lightness or darkness) of the original hue.
To do this with dyes on white yarn, I had to first find a gray that was the same relative value as my saturated color. So I dyed some grays using varying amounts of black dye on the same white yarn. The grayscale version of the photo show that the blue is lighter than any of my trial grays.
So I tried again.This time I think the next lighter gray (labeled F) fits the best.
Then I used mixtures of this gray and my original blue to create the tones of blue.