Agree to start looking at the wiring you can see on the front of the WH first. If all those wires look good, then pop out the little black rocker switch on the bottom left corner (assuming it is a Suburban). The switch or the wiring on the back side of the switch may be the problem. If that all looks good, then you may be having to get to the back side of the WH where the wiring junction box is located. That may or may not require removing the WH depending on access.
Burned out heating element does not typically emit an odor since it is inside the WH.
Be sure power is off before sticking your hands in there.
