It fixed itself. I purchased the thermostat and it arrived 2 days later. Before I had a chance to put it in we baked something on the convection setting. After the oven cooled, the HE light on the burner went out and the cooling fan stopped for the first time in about a week.
That was about a week ago. I've used the burner since and everything is still working fine. After the stove cools, the light goes out and the cooling fan stops as normal.
Before that I tried to get the HE light and cooling fan to go off by turning off power to the stove for a while and then by using the burner. Neither helped - the cooling fan just kept running and the HE light stayed on.