My tool cabinet is 74" tall and has 17 drawers. Straight grained KD Ash. It has no cross supports. All the drawers work perfectly after 20 years. If I built it today I would add a cross support about halfway up for insurance.
Tool Cabinet.JPG
In your case my guess is that there is stuff piled on top of the cases, adding to the problem.
Can you put the cases side by side and attach them, that would help. Then a solid wood top would help with the stuff on top.
Face frames would fix it.
Is there a back on the cases?