Install a new thermistor

I am seeing heat creep into my heat sink and also the E1 printer reset error every now and then. Result is I can print fine for about 1 hr, then the filament in my hot end above the heat break starts to melt, and my extruder cannot push new filament through it.
My theory now is that my thermistor is improperly installed, thus giving faulty temperature readings resulting in my printer thinking the heat break is cooler than it actually is. I recently had to replace my thermistor as I accidentally cut one of the wires.
So, should I install the thermistor while the hot end is hot? What is the best way to keep it in the hole? And how do I know the thermistor is installed properly into the whole?

Before you go to too much trouble, take the fans off your hot end. Remove the heater block with the thermistor and heater still attached. Place a small bowl or shot glass of water on the build surface and lower the hot end, leads facing up, 75% into it. I suggest keeping the leads out of it only because most of the water we use is contaminated and therefor conducts electricity (pure water is actually an insulator - it’s the impurities that make it conductive) so getting the leads wet will throw off the thermistor reading. So, with the leads dry but the heater block mostly in the water, turn on the hot end heater. If the water starts boil but the thermistor is only reading 80C (for example) then your theory that you have a problem with the thermistor is correct. Water absorbs a tremendous amount of heat and will hold the hot end at 100C. The nice thing is that it’s a predictable temperature at standard pressure, so unless you live on a mountain (a real mountain, not Hamilton “mountain”), you can use this to confirm that your thermistor is calibrated.

The fact that your printer appears to be suffering from heat creep can be caused by a few things.

You are correct that excess heat is a possibility, but a secondary symptom of running filament too hot is excessive stringing. Are you experiencing excessive stringing? If so, reduce the hot-end temperature and be suspicious of your thermistor. However, be aware of one other thing: as I have experienced with a roll of filament I have on my printer right now, sometimes the manufacturer lies. My current roll has a label on it that says the filament should be used at temperatures between 230 and 240C when in fact it prints just fine at 190-200C. Running it at those temperatures meant that I was getting excessive stringing, but it also meant that my filament was melting much higher up the heat break than it should have been, kind of like you’re experiencing. In my case, dropping the temperature to 200C made it my new favourite filament. So that could possibly be a problem for you too: being misled into running the filament hotter than it should be.

The other thing that springs to mind is cooling. Is the heat break fan running? Is there anything blocking it? Is there dust/debris on the heat break cooling fins? Anything that reduces or fully or partially restricts airflow will also cause the symptoms you describe.

So, if the fan is running full tilt and it’s vent is unobstructed, and the heat break is clean, and lets say your thermistor passes the 100C test, then what?

Is it possible you’re feeding filament too slowly? The filament itself removes heat from the heat break as it passes through, melts (absorbing energy), and then gets extruded, where it releases the energy. In some cases people like to print slowly if they want a cleaner, more detailed print, but if you go too slow, the heat travels up the filament itself and can lead to a clog.