There's been no medical study that I've seen (and I've looked at many) that indicates the principle pathway of nicotine into your system from vaping. Everything is so far conjecture and guesses. In the only study that mentions it I've found,
http://www.healthnz.co.nz/RuyanCartridgeReport30-Oct-08.pdf
Bottom of pg23 - "Comments: Inhaled nicotine in cigarette smoke is over 98% absorbed, and so the exhaled mist of the e-cigarette is composed of propylene glycol, and probably contains almost no nicotine; and no CO...." The key word there is
Probably. They're making an assumption. Anyone that says "this study I read says you absorb all the nicotine" didn't read the actual papers. Anyone that says study X says you only absorb a fraction of the nicotine you vape, to me, must post the actual papers from the study to back up that claim or it's just heresay and conjecture.
There's much more simple answers to your respiratory distress (medically that's what it is, no matter how little of an irritation it is)
I've experienced that at higher voltage settings, (2.9ohm@4.7v) but the vape tastes really good, and tons of vapor that will linger in the air. At 3.5v, I can inhale it fully without any cough, and the vapor quantity is drastically less. Not much taste either since my tastebuds are still shot. Point being, your kit and settings will determine how harsh the hit is. Also, how much did you used to smoke (how bad were your lungs to begin with) will affect how long it takes for your lungs to get stronger for lack of a better term.
You can puff like it was a cigar, or draw and inhale like a cigarette, no one has definitively said which is the more efficient way for your body to absorb the nicotine. The best chance for your body to absorb the nicotine would be to draw into your mouth, let it cool, where it'll be absorbed by your mouth, inhale, where it could be absorbed by your lungs, then exhale through your nose, where it can also be absorbed by the membranes in your nose.
edit
@Spider , Fill your mouth with vapor. Next you have to conciously like, 'open' the back top of your throat, sort of like the begining of a swallow. Then you compress your tounge up, forcing the vapor up and out through your nasal passages. You'll have to practice that, and after doing it 4/10 times just now, I find it irritating to the top back of my mouth heh.
eh, ok, long winded post over. Short answer, no you don't 'have' to. But if you just vape cigar style and want more nic, then hold it in your mouth longer before you blow it out.