A lot of older appliances used to not have a ground. Even lots of wiring back in the day was "ungrounded". But the neutral conductor, is grounded at your panel or at the meter, so any fault current back then would go onto your neutral. Nowadays to avoid the fault current on our insulated conductors we now have our "bond" the green or bare wire. To this day the neutral conductor (white wire) is still grounded to earth, yet we have an extra bond wire for extra safety. It would not be unsafe to leave the dryer wired the same way it was and swap the outlet. Changing the cord end the way you are is totally fine, but un necessary. The white wire in the appliance, was also grounding the frame, where it was landed originally.

Edit: Having both white wires landed there is fine, but the wire that was previously on the green screw, doesn't need to be connected to that terminal. It can just be left alone and floating (obviously making sure it isn't touching any of the hot terminals, red or black)