Frank,
I am no expert, but, the shaft can not "wag" without some form of fletching on the arrow. Bare shafts will nock plane to the right or to the left.
for a right handed, person increasing weight , if close to spine to begin with, will cause the shafts to nock plane left and hit low, right
If decreasing weight the shaft should plane nock should be right and they will hit left and high.
The weaker the shaft,ie: heavier point will increase the plane to the left.
The reverse is true to a point, depending on rest setup in relationship to center shot..
Then some shelf adjustment out will help a soft shaft, and if possible some removal or getting the shaft closer to the riser will help on a stiff arrow. AGAIN all is for a right handed shooter..
Then your release will change the tears in paper.. so enjoy that ride..