I don't think my system has an emergency electric heating coil - Houston doesn't get that cold. Some winters we only turn on the furnace 5 or 6 times.In your original post you said "I thought maybe the AC air handler..."
Well, unless you live on the Equator, if you have an "AC air handler" then you may have an Emergency Heat coil that comes on if for some reason your "furnace" doesn't when your thermostat is calling for heat.
Best wishes finding the issue.
In my breaker box I have one 20 amp breaker that feeds the air handler - furnace - fan - wouldn't an electric heater coil need its own circuit?
But if I am mistaken - maybe turning on the heat and letting it burn off that dust or whatever it is that causes the smell the first time you turn on the heat every year - would burn it off - and that would solve the issue?