For what it's worth, in tea ceremony you do what most people here have already advised - you rinse only the tines, then leave the chasen on a "chasen shaper" (chasen-naoshi) to dry. Because the chasen-naoshi is designed to allow the chasen to dry with the tines facing down, any water in the neck* of the chasen should drip downward - and of course that part is 100% exposed to air, anyway, so any remaining moisture should quickly evaporate.
*Probably not the right word
