to the op
caution, this may be dangerous
you have read that the 3 prong setup shares the ground and the neutral.
so if this shared ground neutral should have a loose wire, this could cause the dryer frame to become energized.
and perhaps the metal vent pipe is completing the circuit to ground. that may be why you hear and see sparks.
i would not use this appliance again until this is fixed

if you feel you can handle this, do what you suggested, run a heavy wire from the dryer frame to a nearby outlet ground.

after you add a ground , this should make it relatively safe, but get someone to check it before you get shocked or start a fire.

Mark