No. ELSD if equipped is ALWAYS on. Even with Traction Control off, it is still active. There is NO way to turn it off from any button or setting at the drivers disposal. The button turns on "Lock", and I explained that above, that it is clutch based.
Also, AWD/4WD has nothing to do with the ELSD, as that is just the rear. Yes having 4WD/AWD active could prevent less use of the clutches in the ELSD in theory under slippery conditions, because the front wheels are now helping. Most likely not though, because the rears would most likely still be slipping side to side, which is why ELSD would get involved.
As far as the error. Just because you aren't sensing slip, doesn't mean there wasn't any. Actually it is a sign it was doing its job. Might have just gotten hot, and more likely in my opinion, it might have an issue.