I needed to separate the shape off from the rest of the base and then use the ‘option/push pull method’ (hold option key and push pull up from the bottom to create lines through the curved surface).
The drawers are drawn as they would be built: using separate components. Just as would be done in the shop, the drawer fronts start out as thicker blanks onto which I placed curves. Those curves were then used to trim the draw front. The drawers are nested components and can easily be moved in or out as desired.