I think this is an easy fix (at least in my mind) without damaging anything if the drawer is as pictured in sapphire's post. Take out all the wire bails ( they should pull out easily with a little sideways pressure where they are inserted) and store them in a marked container. Glue a small stop (small wood strip) on each side of the back edge of each drawer bottom using white glue. Next, have a woodworker make separate boxes for all of your drawers and set them inside the backless drawers. The stops will stop the inserts from sliding back, the inserts can be removed if necessary, the wire bails can be put back, the stops could be tapped out if needed, and the whole thing would be original again.
With inserts made separately, you could have some custom made with different sized dividers to hold smaller or narrower items , etc.