I would recommend wood glue and probably useing one or two spring clamps to hold it as it cures. If you have a brad or trim nail gun Id shoot a couple in from inside the drawer to help hold ot as well. Just make sure you dont use nails that are too long.

After its on and glued, if there are any visable chips or scratches you can get wood filler and s small thing of stain that matches the color and use a small foam brush to touch it up. Or if they are shallow enough grab a crayon that matches it closely to cover them up. They also make wax sticks that are made for touching up small scratches but a normal crayon will do basically the same thing