On Monday, April 11, 2016 at 8:47:58 AM UTC-5, wrote:

The hardwood brace/support also serves to strengthen
the drawer structure somewhat - joining the drawer-front
to the drawer back at the center.
All the other discussion about just reversing the drawer-bottom
curve, without adding bracing - seems like it might be temporary ?
John T.

Reshaping, only, is not always temporary, if the over correction is sufficient to maintain a flattened result.

I'd agree that extra support would assure further stability. A narrow metal angle iron type bracing, under neath, would help.

I think someone mentioned placing a cross piece/divider inside the drawers, to function similarly. A divider is a good idea, also, and probably wouldn't look unsightly, either.

Sonny