Ok, I have never owned a corn snake, but I can help you with some basics until you get specific answers from some expert corn keepers
First, take a deep breath! Since he was being poorly kept before, all your help and attention will only make things better for him. So thank you for that!
First, a few questions that will help the forum to be of better assistance to you.
1.) How long has the corn been in your care?
2.) Can you post photos of your current setup? (That way we can see lighting, hides, etc)
3.) is your UTH regulated by a thermostat? That is of the utmost importance when it comes to protecting the health of your snake.
4.) How are you measuring temperatures in your enclosure?
I’m regards to the substrate in his mouth, he will get it out on his own. No need to stress you both out with it. You mentioned that he is in shed now but has patches of previous shed on his body? If that is the case, leave him be until it comes off. He should shed any day now. Some snakes just take longer than others and stress can play a factor.
You’ll be able to re-evaluate that situation after you see how this shed goes. As far as heating, is 80 degrees his hot spot? If so, I think it would be best to raise it a few degrees. You do not need to worry about “cooling” his cool end, per se. As long as his enclosure is large enough and his hot spot temperature is correct (and regulated by a thermostat), that will create a natural heat gradient. You are also correct that a UTH will not affect air temperature.