I've checked all the main components except the motor for continuity and they all check out fine.
The AC breaker gets tripped and when flipped back on the dryer works. Dryer breaker has not tripped.
All parts are original except heating element, which is maybe 3-5 years old.
Dries fine when running and will finish the whole cycle. It will only run that one cylce, though, and as far as I know it trips the breaker when the load finishes.
Update 2016-02-12: The breaker labeled dryer is definitely the breaker controlling power to the dryer. Turning it off stops power to the dryer. I'm not sure which part is connected to the AC breaker.
It's too cold right now to turn on the AC so I haven't tested whether it will go off when the AC breaker is off.
I've investigated more and found that the door switch is broken, so it is not always engaged. Could the switch being momentarily engaged and then disengaged cause the issue?