The bitter tastes that develop in both situations can be somewhat alleviated by purging you cigar as you smoke it. When you smoke your cigar most of the essential oils that provide so much of the flavor will burn up while a small amount builds up in the filler leaf creating bitter flavors.
To purge your cigar start by knocking off any ash, put the cigar in your mouth like you are going to puff and instead blow gently back through the cigar. If your cigar is still lit, continue to blow gently through the cigar, rotating it slowly, until you can see a bright orange ring of fire around the the base of the entire coal on the cigar.
If this was a cigar you wish to re-light, again knock all of the loose ash from the end. Apply a flame to the foot the of the cigar and rotate slowly to light evenly. Place the cigar in your mouth and gently blow through the cigar instead of puffing for several seconds.
If you have experienced either of these situations, try purging your cigar next time to enjoy your smoke to the nub.