When I wax I am also a dipper, but if I were to use a brush, I'd not bother to clean it. When it got cool and hard, I'd wrap it as is in plastic to keep clean, and store with the wax. The wax is kept in a pot purchased cheaply at a yard sale and also kept covered.
After being dipped in wax, what else could you ever use the brush for? If you were using different colors of wax either use different dedicated brushes, or dip in boiling water, swish around till the wax melts, and let dry very, very well. I've never done this, but that's what I'd try first.