varnis

When you have an insoluble ppt, you have separated one ion from the other, in the sense that the Na+ is left in solution. So filtering the ppt will give a solid and a solution. As for the last part of edit #2: No, you can leave the remaining ions either dissociated (Na+(aq) + NO3-(aq)) or as sodium nitrate (NaNO3(aq))

If the above is too long for you, you could consider then CN route, adding sodium cyanide, though you would want to be careful of HCN gas. You can easily yield a Carboxylic Acid of one carbon longer than the starting alcohol in this way (Via a Halogen substituent)