Hi Koen,
I will allow myself to give a bit of advice. Do you play with the drone "turned on"? If so, that may add to the instability you are talking about because, one need a bit of practice and experience to keep an even, unchanging tone playing with the drone. Here in Bulgaria, those who start the kabapagpipe and have no previous experience with such instruments play about six months without the drone.
In short, check the chanter without the drone on some occasion.
It is perfectly normal for a new reed to go high. In course of time the the grains of the reed will get more and more lax and the reed will play lower. For now much more important as for tuning is not to have the bagpipe exactly in D but to keep the notes/holes intact.
Finally, If you have time make a video and send it to me. Thus, it's gonna be much easier for me to give a valuable piece of advice.
Keep on and don't despair,
Kostadin