From what I can see in the picture, it looks to me like the drawers are designed to slide upon rails which are surface-mounted.
Your pictures do not make it clear why the distance between the left and right sides is now too great for the drawers to sit atop the rails. The horizontal and inner vertical planes of those rails appear to be pretty sharp and even, so it’s not looking like repetitive use has worn them down to the point that they are slanted and allowing the drawers to fall. Instead it would appear that the left and right sides of the armoir have been allowed to move apart. Until the cause of that is stopped, smaller repairs will be worthless.
Off-hand I might suspect that the glue holding the larger framework together has dried out over time and the sides are no longer tight. However, I’d have to inspect more closely to know for sure and I’m too many miles away to just drop by for a quick glance. I can’t ascertain whether the top of picture #2 (the close-up) is showing the front edge of a ring tray or the front of a horizontal ‘bar’ of wood. If the former, then the sides of the armoir are attached to the piece of wood above and their attachment points are what keeps them from spreading further apart. If the latter, then it’s possible the sides of the armoir also attach to that bar in the front, either via screws or via dowels (quite possibly with glue, as well).
You might be able to open the doors on the sides (necklace racks, right?) and see if there are screws (or holes where screws used to be) for attaching the sides of the armoir to that wooden bar in front.
You might also try to ascertain whether the vertical sides of the amoir have become warped (bowed outward). This could occur after long periods of moisture in the area, with or without heavy weight pushing downward on the top of the armoir. If the sides are warped, your best bet is to replace it. I believe they still sell similar units at JC Penney.
Best case scenario:
[li]The sides are not warped or bowed. [/li][li]The horizontal piece at the top of picture #2 is a connecting piece rather than a ring tray.[/li][/ul]
You can either replace the lost screws or drill & counterbore new holes in the sides to go into the top connecting piece, then use new screws to attach the sides more firmly to the connecting piece. That may be all you need; just restabilizing the sides so that the rails align at the right heights and distances apart.
If it was mine (my wife’s) I’d replace the rails with wooden pieces that are equal in depth into the armoir, thicker coming away from the walls upon which they are mounted and shorter in height as you look at them – by about 1/8" so I could mount a MDF board (or pegboard) smooth-side-up atop each pair with countersunk screws. Then each of my ‘replacement shelves’ would go into the armoir to replace pairs of rails so that the drawers would ride on the MDF board and each shelf would attach (via countersunk screws) to the sides to help maintain the distance between the sides.