Assuming that model number ends in EG7WW (not EG&WW), the problem with the motor not running unless you hold the start switch indicates either a faulty idler switch or faulty motor. The burning during heat cycle could be any number of things. Do you have a multimeter and experience using it? If so, check continuity on the idler switch, also known as the belt switch. The arm of the belt idler pulley activates this switch. The switch is normally closed as long as the belt is intact. If the belt breaks, the idler arm releases the switch which opens and turns the motor off. The start switch bypasses the idler switch. Also check continuity of the start switch between terminals 4 and 3 (two brown wires).

Eric