I cook birds at ~300 for around 1 1/2 hours. I cut out the backbone and seperate at the breastbone to create chicken halves that lay on the grill.
The last part to get done is the joint between the thigh and breast. When I think the bird is about done I seperate them while I'm still outside. If visual inspection reveals that they need a bit more time, I put them back on the WSM with the undercooked part facing outside and the cooked part facing inside.
Another trick to help make the skin edible. Take the chicken off the grill with about 15 minutes to go and use a propane blowtorch to crisp the skin, then put the chicken back on the grill. I get the torch really close to the skin and move fast. The skin will retract as it crisps.