I'm really new to this, But I had the same trouble with my Brinlmann Smoke King Deluxe, I made a couple of modifications, seemed to help. First I covered the floor of the fire box with fire brick, then I threw the worthless charcoal grate away and welded up a basket. (temporary one until I can afford on of Al's. I also put some firebrick in the bottom of the smoke chamber, Not fitted like the fire box just laid in for kind of a heat sink. Oh yeah I had to put legs on the basket to raise it up a little. Also made a big difference. After my disaster with the whole chickens, I now bring it up to temp, 250-260 with Kingsford and then use lump the rest of the way. Seems I have to feed it a little more often but the temp stays up and the flavor seems to be better.
I don't care what those people in Wisconsin say Heaven on a bun is a pulled pork sammich!!