In answer to your question, yet, you are safe. My smoker is never plugged into a gfci outlet. I just make sure I am wearing rubber soled shoes and not standing in water.
It could also be the gfci as they do wear out. You can plug in a heat generating item into the gfci and see if it trips it. Check the amp rating on the smoker and make sure the item - toaster, toaster oven, steam iron, etc pulls the same or greater amps. If the outlet is bad, it SHOULD trip but no guarantee. If it trips, replace it. If it does not trip, move on: Plug your smoker into another gfci outlet, preferably on another circuit (check your breaker panel). If it does not trip, the other outlet is bad. If it trips, test the outlet with an heat generating item as previously described.
There are lots of things that can be wrong up to and including a short to ground in the smoker which is easy to check with a multimeter. A short to ground is through the smoker to you with you being pretty much the ground. Thus, the use of a multimeter and the smoker unplugged. You or a friend can handle this and make the correction. You may have a loose connection in the smoker and again that is an easy fix.
As a word of caution: I am a tinkerer, not an election. Between books and youtube I manage to do a lot in a safe manner.
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