Plastic doesn't melt until it gets pretty hot but any heat can make it softer and possibly allow stresses in the material to deform it.
How are you measuring the temp of the uth? You should be measuring the temp of the glass that's right above the uth. An infrared temp gun is best for that. That way you don't rely on what the thermostat actually says, the numbers it shows are just a scale to adjust the actual temperature you measure.
Now the ambient temp that you're using a che to get you need to have the probe hanging. I like to have it on a wall of the cage right next to the che but about halfway down the wall. You will need the thermometer probe somewhere away from that a little to get the ambient temp.
The temp of the glass over the uth should be about 90f.
The ambient temp should be around 80f. And you should check the ambient temp in a few places.