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It would help massively if you find yourself sitting a bit too far forward with all your body weight on your hand instead of evenly distributed between the saddle and bar (you'll notice this when you let go of the bar and find yourself pushing the pedals harder to keep your body upright).
Work out this first before you spend money on everything else, once you got your saddle position in an idea position you like, then you can tweak the stem length.
A quick, easy and free tweak is to drop the tyres pressure on both front and rear by 10psi.
What's the maximum tyres you can fit in that frame? rear 28mm no problem, but the front if you can fit at least 25mm, you should able to reduced the amount of vibration from the fork.
Also a slightly padded bar tape might help.