Just use the smoke tube in the smoker box. I do that all the time. Cold here as well so give it three to four hours for a nice smoke. I do the same thing in the summer time. The smoke tube produces plenty of smoke for what you’re doing.
Mine has done this several times and seems to be a combination of residual heat in the fire pot and the pellet prime at the end of the burn down cycle. I agree the prime should happen at the start up, not the shut down, but not even a big deal in my opinion. Smoke on!