I believe the problem is you're not using a thermostat to control the UTH. You never EVER use a UTH without plugging it into a thermostat to control the amount of heat it puts out.

They need to be controlled. They can not just be plugged in by them selves and left to go full blast.

I'm sure you do need the larger one. You just need the proper set up with a thermostat.
Then you need a digital thermometer with probe to measure your hot and cool sides. These are very cheap to buy at Walmart.

The vet sounds questionable. You should have been given silver sulfadiazine cream for the burn to aid in healing and prevent infection.

I wish you luck with your little one. If you don't fix the problems with it's enclosure, I'm afraid you might run into problems later on as well. Your snake could stop eating more often, get sick with RI, or run into digestive issues due to not being warm enough.

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