I have the Itasca detailed below, and lately has developed an issue where the dash fan will only work on the high (noisy!) switch setting. After starting the engine (Chevy workhorse 8.1 engine chassis...
Your resistor doesn't "look" bad and the heat marks on the coil are normal as it's burning off energy instead of spinning the blower fast. You need to find the schematic or diagram for your chassis.