Am I doing this wrong?
No, you just do not understand how these devices work.
No biggie, hopefully I will be able to explain it.
The thermostat should be 0 ohms at room temperature and infinite ohms (open) when they reach the set temp.
So lets say a thermostat has a set temp of 200 degrees F, current will pass through it till the heater and the thermostat reach the 200 degrees and then it will open cutting off the current and stopping the heating. So the heat is regulated at 200 degrees.
Another thing to note is that when current passes through a resistance it causes heat. This is how your heater coil works, as an example a 10 ohm heat coil with 240 volts across it will pass 24 amps of current. This is then a 5760 watt heater (current X voltage = wattage)
So the resistance in the old thermostat acts like a heater when current passes through it and that is why it heats up/chars when it should not.
The new thermostat has no or minimal resistance so no heat is generated as the current for the heater passes through it and you only get heat at the heating coil.
Hope I explained this OK