Yes you can do a lot without refinishing and do it inexpensively and quickly. The conditions of storage apparently did not allow for sufficient ventilation or temperature control. The stress caused by changes in atmospheric conditions have caused some structural damage which can be aborted with Wood Elixir but not reversed. The finish problems can be solved with Wood Elixir including the mold, finger prints and whatever the sticky stuff is will be substantially reduced or removed with one or two treatments. You can do it yourself with no special skill required. The oil products you refer to are mineral oil based (like lemon oil) and and therefore will not dry and in fact serve no useful purpose. Where the finish is loose and chipping off can be stopped but the color will have to be touched up with lacquer dye markers.