I confess that I trimmed the top of the door at a very slight angle and moved the hinges. I did try shimming under legs but never got a satisfactory result.
The back of this piece is frame and panel, with three 5/8” thick cherry panels. My opinion is that the design of the front framing does not provide enough rigidity. I reserve the right to label that opinion as complete bunk after I gain more experience with larger, more complicated pieces like this.