Trying to understand wiring to old heat lamp in bathroom
Since you are on breakers 10 &amp; 12 it is probably a multiwire branch circuit sharing the neutral, today's code requires handle ties if this is the case. Verify the hot feeding the light and switch. A noncontact tester may show a hot line or a phantom induced voltage. Depending on the wiring it could be a switched neutral this would be a code violation and is common in home owners doing there own wiring. The correct way would be to take the black wire that feeds the light and connect it to the black that goes to the switch, in the switch box the white wire that comes in with the black needs to be marked with paint or electrical tape a color other than white or green. The white in the light on the same cable feeding the switch needs to me marked also. Heat lamps usually take special fixtures that have high temp wires and many of the heat lamp fixtures have a maximum wattage of 300w some much less. Excess wattage lamps can be the cause for the discoloration of the insulation. The wiring needs to be evaluated and possibly repaired with proper junction boxes added. It would be a good idea to add handle ties to breaker 10-12 if it is a multiwire branch circuit. They may not have been required when the home was wired but are a good safety feature now. This or replacing the 2 breakers with a double pole breaker with 1 handle.