Mostly sounds like you have non-uniform temperatures in the mash and need to thoroughly mix it. You're just measuring two different temp zones. An igloo cooler should not be losing that much heat. Don't worry too much spending a couple minutes just stirring and breaking up the grain. Even with the lid off, it won't lose much heat over a couple minutes.
Or like you said, could be the thermometer. One of mine had a a plastic/rubber dipped tip, and once that got slightly chipped off, it started reading all over the place.