I noticed a big difference in this area when I started drying the baccy more prior to loading and smoking. This may be a little unorthodox, but it works for me... Try drying it out almost completely, say 30-45 min. Pack your pipe, then place your lips around the bowl and put a few long breaths of air through the pipe. The baccy will pull some moisture from your breath. For me, this is just about the perfect dryness. And I get to the bottom, without bad tastes. But it may still take a light or two after a tamp when you get down there.
Good luck!
Have fun experimenting, it comes with the hobby!