From your description of the cable it sounds like it is type NM cable. If that is the case this installation was never code compliant. The bare wire in type NM cable is not to be used as a grounded conductor. I agree with Bob that you haven’t made this installation any safer and have made the problem worse. The frame of the dryer is not grounded at all now. I recommend getting a qualified electrician to replace the cable and install the 4-wire receptacle.
Curt
