Before I did anything else, I would take the motor out of that blower housing and take it down to any electric motor rewinding shop.

I'm thinking the high amperage draw on high speed that's likely causing the black wire to heat up is caused by friction in the motor itself. Also, I'd clean up that impeller if it's dirty.

The rewinding shop will bench test the motor to see what current it draws on each of it's different speeds, and those results will tell you if the motor has high internal friction.