I have seen this type of wire, and yes, it was popular in the early 1980s.
You can run an external grounding conductor, but it has to go back to the panel, or tie into your grounding electrode system. It is not permissible to connect it to an adjacent circuit's ground wire. [see NEC 250.130(C)]
But if you are going to all that trouble, you could simply replace the 10/3 with a 10/3g cable.
But you should be aware that the Code does allow for 3-conductor outlets for dryers, with grounding thru the neutral conductor in existing installations. All you have to do is re-install the bonding jumper in your appliance, and use a 3-wire cord in your existing receptacle.
I think that will be the simplest solution. I hate to not have it set up right, but I don't hate it enough to have to rewire it. The cable runs from downstairs to the attic and back down to the panel. Life would be so much easier if the lowest common denominator was not the highest so many attempt to achieve.