Take the drawer out and check the pieces it sits on. They may have come loose and the extra weight pulls them down when drawer is open. You may be able to use glue to put it back in place or it might need a very small thin nail from the outside. Look for finishing nails in a color close to the finish of the furniture or you may be able to put a dab of paint to hide it. Get shortest possible so as not to go all the way through slide and stick out inside.
Is it the bottom of the drawer that is sagging? If so, you can take it out, turn it over and it will be o.k.
Hitching on Deanna’s answer, you may be able to slide the bottom out of the drawer and flip it over. Reinsert the bottom with a little white glue and you may get years more service from the drawer while doing nothing else to it. Jewell
All of the above are good answers, You might also have a 1/8"-1/4" piece of plywood cut to replace the bottom, and glue into the drawer slides and front plate. You could also add an extra cross bar to support the bottom by getting some flat wood trim and notching the center piece to accomodate the depth of the new cross bar. Lastly, stop putting so much weight in the drawer.