Aw, Bob, I feel your pain. I've learned the hard way to always hold onto the phal if I turn it over. Perhaps you're one of the lucky ones and it will grace you with another inflorescence.
Ryan's use of a straw is briliiant. And, I think I read somewhere that Peter Lin uses his wife's hair dryer to blow out water from the crowns of his giganteas.
I use the paper towel method also. I stick the corner in the crown and it absorbs the water. That way I don't even have to touch the pot or the orchid. Good luck.