Sounds like you've already taken care of the first step, eliminate the extension cord and receptacle as a possible reason. If you plug your smoker directly into a functioning receptacle and it trips the GFCI or breaker the most likely cause is a short somewhere in the electrical circuit of your smoker.
I'd recommend first visually inspecting the cord and all visible wiring for worn insulation, severe twists and kinks or exposed copper. Insure the connection is dry as well, a wet plug can trip your GFCI (as you probably are well aware)
Sounds like you were already checking continuity with a meter, a good check for the heating element is checking continuity between the beginning and end connection of the element. You should see pretty low resistance there indicating no breaks in the heating element. Make sure there is no contact between any of the metal of those connections and the frame/body of the smoker. Sometimes a connection can come loose there and fall down and make contact with the smoker, that would cause a short.
If you can't find the problem visually my advice runs out at that point. Not sure what kind of thermostat you have on it, the short could be internal to that as well. I don't know this particular smoker and don't have an electric one myself but 15 years of building and servicing electrical signs left me with enough knowledge to get in trouble.
Hope it helps a little, good luck! I'm currently trying to repair my propane smoker which I'm not doing very good with. Smoker's down make for a very unhappy cook.