Hi Johnny;
Welcome to the forum. Crisping the skin in the Bradley, you may not get good results, even if you get the cabinet up to 300°F; due to the amount of moisture that will be inside the cabinet. If you can smoke/cook the chicken close to 250°F from beginning to end, though the skin will not crisp up at that temperature the fat under the skin should have fully rendered out, and the skin will be bite through. The best recommended way to crisp up the skin is to finish them in the oven at high temperature, under the broiler, or on the grill.
The earlier models that switch would not disengage until you got over or very close to the 320°F mark or a little over, and it seems you have an earlier model. Those switches have a range in which they could disengage, and could disengage after it when over 320°F, or as low as 250°F. Then later Bradley tighten range, and it would disengage at a lower temperature. Your switch may be off a bit, and possibly replacing it, you may get a switch that will not disengage until a higher temperature is reached. But that is a crap shoot.
Not recommended, but since you have a PID you can rewire and by-pass the switch. But if your PID should fail, you don't have any safety backups, except the inline fuse.