re: Smoker is rolling

Dang. I hadn't even considered that. I will need to do some research to see if that might be it.

ETA a couple things I found.

You can go low and slow, but the skin will absorb LOTS of smoke and usually get rather rubbery. I usually smoke my chicken's much higher as @Sean Hart suggests and have great luck there. Low and slow works best with cuts of meat that are particularly fatty - the meat in chicken isn't especially and doesn't really gain that much from it. If you go low and slow, you need to add moisture and/or fat to help the meat out. Additionally, since the skin absorbs smoke very easily it, depending on your wood, may become bitter with smoke.

And another:

Cooking at really low temperatures can result in a rubbery skin. Chicken is one meat where I've never seen the need to go low/slow. Brine the bird if you're worried about drying it out or if you want to get some extra flavor into the meat, but it's not necessary. In any event, make sure you get the skin dry before cooking. Overnight in the fridge should do the trick. I like to rub the skin with vegetable oil and BBQ rub (or sometimes just salt & pepper) and cook it at around 300-350f on the smoker. Not only does it cook faster, it also doesn't take on an overwhelming amount of smoke.
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So I was kind of guilty on both of those fronts. I probably hit it with too much smoke and I didn't dry the skin out before throwing it on. It was out of the brine, rub, and into the smoker. Others are saying that tossing it on the grill or in the oven at higher heat will help mitigate rubbery skin.