[NightKnight]

Try adding some weight in the front.

 

[NightKnight]

Here is an image from that Aerodart site. Note how the Model Dart has no nose waggle, whereas the dart with a heavy tail has nose waggle.

 

[NightKnight]

So, you basically want to bring the mass forward on the shaft, causing less waggle.

 

[NightKnight]

The throwing dart guys have put a lot of thought into this. That is why they use a heavy metal in the front, and aluminum back by the fins. Now, our fins/flights (cones) aren't nearly as effective as those on a throwing dart since cones dont provide a great deal of directional stability. So, front mass can be even more important. Again, this is all IMO.

 

[NightKnight]

Here is a pro dart. This shows how the weight is balanced better.

 

[NightKnight]

So we figured out the wobble, but why the veering off of course after about 10m? It is very strange, they get to about 10m and then they take a 90 degree turn left or right

The aerodynamics may be such that, after it slows down to a certain speed, the mass of the dart may not be enough to keep it on track. The weight issue is also involved here as well.

 

[NightKnight]

Why don't you try taping a nail (of similar diameter) to the nose of your bamboo dart?