You can also get a safe circuit by using that old wire and installing GFCI receptacles at the first receptacle location on each circuit.
You can't ground the old circuits by just running a ground wire from the device to the box, because it won't be grounded back to the panel. It will not have any effect.
You will probably be shocked at the cost of running new wire and may decide to use the old with GFCI protection. It is hard to run new circuits if you don't have access above and/or below the walls where the receptacles are located.