Sean, It will depend on what color your sock is now and which color you choose to dye it with. (red+green = brown, purple+yellow=brown, orange+blue=brown) <--okay that's pretty simple, but you get the idea.
Make sure you get enough kool-aid to make a real strong dye.
You can get a more variegated color dying the yarn when it is dry (or part dry, part damp & part saturated with water)
If the original hand painted yarn bled that badly, then the dye in it wasn't set very well or it wasn't rinsed very well (i.e. bad dye job)