Repacked packing nut but sink is still leaking!
Is this the cutoff valve for the cold water supply to the washing machine? Does it take several turns to go from closed to full open on the stop?

If, when the valve is closed, it leaks continuously around the stem, then the plastic washer at the bottom of the stem needs replacing. One has to remove the stem to do this. Some of the valves of this type use the packing nut to cover the interior and other valves have a separate screw cover. These latter can be stuck in place, but if not stuck this is a standard maintenance procedure.

With the stem out to change the washer, one inspects the brass seat in the valve body. If the sealing surface is compromised, then it should be replaced. However, a valve in cut-off service for a washing machine is hardly ever cycled so the seat is probably OK.