The purists will be aghast, but you might be able to mask some of the remaining stains /streaks with a bit of pigmented oil brushed in carefully as the first finishing step. Don't use water or alcohol stain, because they are generally dye stains. You want a pigmented stain, meaning one that leaves a sludge in the bottom of the can after it sits a while. The water stains in your stock will generally follow the grain lines in the wood, so you could use a striping brush (sword brush a used for automotive pin striping) to carefully color the streaks and blend them into the rest of the wood. If you screw this up, it will be hard to correct, so be very careful. Start with a stain that is lighter than you think you need. You can always go darker but it's hard to lighten things up once the pigment is in the wood.
Mark Larson should probably comment on this one.