Through the neutral, the center lead from the cord would just bond to the case.If you use a 10-30 how will the dryer be bonded?
That's probably what I will do. Normally Id just pull a 10/3 with ground but the laundry is on the second floor with the panel on the other side in the attached garage. A new home run would be impossible.It it is fed from the main panel, I'd leave the 10-30. The white is already landed on the neutral/ground bus.
If it is fed from a sub-panel, I'd change it to the 6-30 and re-land the white on the ground bus. And tape it green.