The underlying problem is that the tracks the doors sit in isn't strong enough. You could address the symptoms by adding catches or friction or weights.
Fix the cause - strengthen the top panel of the drawers. This will take away the sag and keep the track level. Also allows you to sit here to put on shoes or similar when door is open.
First, decide if you want to add the bracing on top of the drawer cabinet where it will be visible on opening sliding doors, OR add bracing inside the drawer cabinets which will impact how much you can store inside each drawer.
For an invisible fix, remove the top row of drawers and add dwangs/battens right under where the track travels. You need to glue+screw the battens in place and probably into the sides of the drawer cabinet too. I'd aim for a minimum height of 12-18mm and more would be better. You could have a greater depth, perhaps 50-80mm.
One could screw a metal angle extrusion underneath, if woodworking is not your area.
If the loss of height in the drawers is too great, you could re-work them to be single layer drawers with a false double-height front. That would let you use most of the drawer space for storage.
The visible solution would lay a single length of timber across the top of all the cabinets, right behind the track This will stiffen the top of the drawer cabinet to reduce that sag. Since its visible, you need to think more about finish too.
I'd be tempted to install a line of LEDs behind too, to help illuminate the contents but that's just me.
A third option is to re-panel your sliding doors in something much lighter. Your photo suggests they are faced with a wood panel, which is likely quite heavy. Consider taking them off and redoing the panel with cloth stretched in the frame, or a Shoji-style of Japanese paper panel. Clearly neither of these are structural, so the frame needs to have reinforced corners but that could end up with a lighter panel that won't cause the drawers to sag.
Last resort is to do away with the doors completely and have exposed contents. That may not suit your interior style, and does place your clothes on display all the time, but makes the room feel slightly larger.
There's a remote chance its possible to find a stouter bottom track that will resist flex more, and therefore hold the existing heavy doors, but that's going to be awkward and may require "shortening" the sliding doors - which sounds like a lot of pain.