Like they asked, wood or charcoal, and are you using lit, ashed over coals? There will always be some thick smoke as charcoal/wood lights, requiring some finesse. Add lit coals, and when adding wood splits, keep the firebox open for a few minutes to "burn off". Some people don't start until after burning down a few splits to a bed of coals, then slowly/carefully adding them.
Those vertical offsets are designed to be efficient, and it sounds like yours is. You could use water on the bottom rack as a ~225*F heat sink til you figure it out, but you definitely do not want too much thick smoke.
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