Can the booster appear in my cart drawer? (Essential Free Shipping Upsell app)

Yes, in supported cart drawers. There is no separate cart drawer booster type. A Cart page booster can also be shown in a cart drawer that the app recognises.

The app recognises several standard theme cart drawers, the Prestige theme's drawer, and the Essential Cart Drawer. Recognition depends on the drawer's structure, so a custom drawer may not be recognised. For a drawer that is not recognised, choose Custom position and use the booster's app block or code snippet. For a third-party drawer, the snippet may need to go into the drawer's template. Check the result in your theme after you add it.

Cart page boosters are available with the Essential plan. Your plan details are shown in the app.