81-83F won't burn your snake, but, if it feels hot to you, that indicates that you have temps that are hot enough to burn. How are you measuring your temps? Is it the surface temps that feel hot or the ambient?
I'm a little unclear on how you are heating your enclosure. Do I understand correctly that you are using a CHE to heat the warm side and that the cool side isn't currently being heated due to a failed thermostat? What were you using to heat the cool side, a CHE as well?
Pics of your enclosure might help us advise you better. Posting pics on Imgur works fine. If you want your pics to appear in your posts, you can just drag and drop from Imgur. That's how I do it.