The probe needs to be outside the cage. Only bad things will happen when controlling uth with the probe inside the cage.
The snake CAN get under the carpet. It is no protection.
You need to know your temps. Get an IR temp gun on Amazon, they are cheap.
If it were me I'd cut the tstat down to 88F and get a IR thermometer.
The uth does not heat the air and it is improper to attempt to control the air temp with them.
I don't know your tstat but if it's a cheap one that says on/off and that it's not proportional it does just as you said and turns off and on to control the uth. Proportional tstats are best and one day you'll want to upgrade.
So much for backing out. I think some full pics of your cage would help me get you up and running. One trick to make a hide feel smaller to your snake is placing a good amount of moss in it covering. This covers the large opening and takes up space inside making it feel more secure.