The dishwasher does not start when it is supposed to, but can be started by spinning the motor wit a wooden spoon. I don’t see a start capacitor in the parts guide so I assume it’s in the motor and I need a new motor?
Here are your parts
Replacement parts for GE GSD1200G00WW | AppliancePartsPros.com
If the unit once running sounds OK then I would say you need a new Motor & Pump assembly.
Check the condition of the dishwasher to determine if it is worth putting the dollars into it or would you be better off with a new unit.