One idea is two layers, with an inner tabbed box supporting the middle shelf and separators and an outer clean box glued around it. Your lid could then be two layers, the bottom smaller than the top to provide a secure lip. The pictured example just uses a single layer top.
(That’s box #2 from Box-A-Day Challenge)
For the drawer, it might be easiest to move the rails to the outer edge, simply extending the left and right sides downward to form them. A matching slot through the outer box and tracks on the bottom of the inner box would complete things. A vague/crude sketch:
(Literally on the back of an envelope.)
On second thought, I wasn’t thinking all the way through… The face of the drawer needs to be clean, so the outer box wouldn’t have slots at all, just a rectangular opening. The face of the drawer would extend down to be even with the bottom of the tracks.
Sorry, thought I’d jot this down while having my coffee, but it’s hard to convey such things concisely.