I HIGHLY doubt you are going fast enough to necessitate a stiffer board. I'm willing to bet you have way too much weight on your back foot.
Try to consciously quantify what percentage of your weight is on each foot as you make your way down the hill. Feel the pressure on each foot and aim for 50/50. This will feel different on the hill from how it feels on flat ground. Since you are on a hill/incline, 50% pressure on each foot will feel like you are leaning down the hill (when your snowboard is pointing down hill). Just remember this as you try to equalize the weight on each foot.
The time when everyone goes in the back seat/excess back foot pressure is usually just as they initiate a turn, and the nose starts to point downhill. At this point you pick up speed, and it is very natural for a novice to put more weight on their back foot. This takes weight off of your front foot and takes pressure off the nose end of your engaged edge. And here it is---> this release of pressure from your front engaged edge is reducing the hold of your board on the snow and causing the flapping/waggling feeling.
In reality, 50/50 weight actually "feels" like 60(front)/40(back) since you are going downhill. Aim for the correct weight distribution as you make you way down the hill and I think you will have a better outcome.