Greetings,
Since this deals with the light/turn signal, it sounds like you would need to unscrew that portion of your scooter and the relay should be present there. You can use an image from ebay to help you identify it once you've unscrewed the covering for your flasher/blinker.
Thank you.
There is nothing that you can do. The motor and gears are designed for the best performance. If you change them you loose on the other end of the spectrum. IF you add batteries you add weight and may not have space to add more.
Melted means overload, blown means a short, possible causes-bad connection on battery leads, motor brushes worn, shorted wiring in motor or bad bearing, brake dragging on wheel, motor controller going bad. Check it with wheels off the ground , try to isolate what is binding , causing overload on the motor.