I'm quitting cigarettes, not cigars. I inhale vape just like a cigarette, because it's designed to meet that need (hence, throat hit). If you want to stir vape around in your mouth then snort it out without inhaling, more power to ya, but my habit could never be satisfied with that method.
Sounds like you are taking the vape into your lungs... that is not the "correct" procedure.
Most of the flavor and nicotine will absorb in the mucus membranes of your mouth and nose.
There is directions on the proper way to vape somewhere ... if you feel up to looking for it (reasons and stuff included)
Basically, quick hit to warm up the vape - the long slow hit - hold the vape in your mouth - roll it around like a cigar or wine -
exhale through your nose.
No reason at all for that vape to be hitting your lungs.
Menthol might be soothing to your throat with a chest cold and all.
Good Luck