I've been told that the drowsiness does wear off after a little while. I don't think I had a problem with that at first so I can't tell you any more than that!
As for trying to get the water to flow out the other side, it hasn't been a consideration for me at all. Most of it I spit out or it goes back out the same nostril. I know I've read things from other people saying that it works better for them if they don't worry about it going out the opposite nostril. You could try using a big spoon and just letting the water come out however? Just so long as it all comes out
Let me know what's going on!