4 hrs isn't close to time, depending on temp. You need to wrap a few bricks and place them around the base of the cooker. Too much smoke flavor is too much wood used. Try more charcoal and wood sparingly. The wood / smoke is secondary to maintaining heat with charcoal, propane or electric.
Grab one of these they are great and do not leave fibers over the grill like a fiberglass blanket would. I picked up the 3x3.
http://store.cyberweld.com/panfelwelbla.html
IF you had a wind break, the temp would maintain more consistent over the cook time. For the winter our's is on the deck and have a 8x6 canvas for wind break AND to minimize possible fire. The real reason i have it is to keep the spouses concerns about fire down, as the grill inst suggest a good 3' + from any combustible material, like a wood deck.
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Mesquite is one of the strongest woods for flavoring meat. If you're not careful, it can be downright bitter.
The other concern is that "smoking" is not necessarily "cooking" depending on temps.
I like apple wood for milder flavors. I also hear alder is a good subtle wood across the board, but I've not used it.
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