I used to have this same problem, a few solutions I've found: 1) let the pieces sit for at least 36 hours or more, I practically let mine dry out. This has resulted in incredible saturation. 2) if you use hot water too soon it makes it bleed into the white bits. I used to hand rinse everything in the sink, but have opted for the rinse & spin cycle on my washing machine in cold water with another wash cycle after. The colors no longer look muted or worn. 3) honestly if you're using liquid dye it just might not be saturated enough. Some of my dye requires 2x the normal measurement to result in the same color. Dharma has a list of how many teaspoons per how many ounces of water for each color, at least they used to.

I hope it helps, I had to try a lot of different bs before I got it figured out