It sounds like tinsel lead slap to me like this
it is a fairly easy fix. Take the subwoofer out and put something soft like foam insulation or speaker gasket tape between the leads and the cone. Or they have terminals that can be turned or bent. Every sub will be made a little different so it is hard to tell you what to do without seeing it.
I would say the reason it stops when you touch the cone is because it limits the excursion just enough so that the tinsels are not able to hit the cone and vibrate.
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