Is it possible to apply a shipping cost that is product specific?

I need to apply a shipping cost to cabinets differently than drawer boxes as well as doors

The best way to do this would be to add tags to each of your products that you want priced differently then you can do a price adjustment. So you can tag all drawer boxes as "drawerboxes" and cabinets as "cabinets" etc.

Once you have tagged your products go to Settings -> Operations -> Price adjustments. Here you can select "use tags" and enter in which tag you want to use as well as the price the adjustment you want to make. Be sure in the "accumulate" column that you select "always" so that the shipping is applied every time. Here is an image of the price adjustment page for reference!