Although I don't do this, I would think for easy access you could use muffin tins in a drawer. You could keep a spool with its bobbin in each cup, see them at a glance, not even have to take the whole bin out of the dresser drawer . Of course if you DO want to move it all it once, you'd have to carry it carefully.
What about styrofoam egg cartons? Again, not something I've tried, but you'd have separate compartments for small spools and bobbins, with lid, in this "case."