One word: DRAFT
A stick burner HAS to have a strong draft.
Compare it to the fireplace in your living room. If you were to close off the damper and light a big fire, what is going to happen?
Everything in your living room will get covered in soot/ashes, and your house will fill with smoke.
Same thing happens in the cook chamber if the air flow is weak.
The biggest contributor I found to poor draft is a leaky firebox. (assuming exhaust vent is open 100% of course).
Fire needs oxygen to survive, and like everything else, it will pull air from any source it can find.
If the firebox is not sealed well, then the fire can pull smoke from the gaps around the lid, between the firebox and the cook chamber or anywhere else it finds an air gap.
So now you have a fire going strong, and smoke (TBS or otherwise) coming from the exhaust.....however if the fire is ALSO pulling air from around the fire box, the air flow (draft) through the cook chamber will be low. If the draft is weak, soot & ashes will land on your food as it lazily drifts by.