On the woodstoves I have, I always put a damper in the flue pipe just above the stove. When it gets huffing like that, its burning faster than the air supply coming in can accommodate. You need a damper in the flue to balance what's coming in with what can escape. In other words, make the "out" hole smaller in relation to the "in" hole. Sort of like balancing saw muffler mods with the right intake/carburetor changes or tuning. That will settle things down on a stove that's burning that hot. It will also help hold the heat in the stove and make the fire last longer, but if you choke it down too much, especially if your wood is wet like you say, then you will create a creosote problem in your flue. The insulated pipe will help that some, but as you say it just takes some time and experience to sort things out and see what works best for your setup. The stoves in my house and workshop are very similar old-school design, venting into identical masonry flues of almost equal height, but they both have to be managed differently as far as loading and air flow.
If you stove has a built-in flue damper that I missed, or if you already know all this, then I apologize. This is offered in the spirit of help and not criticism.