I didn't think the capacitor had anything to do with the fan motor. I thought they were for kick starting the compressor only. I could be wrong but that capacitor does look shot.
The way I understand it, there are 2 capacitance in that capacitor in the pic.
The high capacitance run the compressor and the low capacitance run the fan motor.
This is in the simplest system.
To OP, wash the coil also if it has not been suggested above.
Turn of the breaker switch to the outside unit before doing so.
Then, blow the condensate drain pipes in the inside unit.
You can make a vinegar solutions to rinse those condensate pipe.
If you don't have the ability to blow, call an AC person and they have the nitrogen tank to blow the pipe and do proper maintenance.
Replace the filter regularly otherwise the evaporator coil will plugged up with dust.
Use pleated filter instead of fiber glass (cheap) filter.