You say the outside unit is running. I think I and others in here are wondering if you are just talking about the fan on the outside unit and not the compressor.
Anyway, simple to check if you dont know what we are saying. Feel the low pressure side of the pipe going into the outside unit. The lower pressure pipe/tube is the big one.
The high pressure side is the thin one.
The low pressure side should be cold and sweating in most cases, if it feels air temperature warm chances are the compressor isnt running. Whether you are handy enough dont know but replace the Capacitor and the Contactor both cheap parts less that $100 for both.
If you are unsure what you are doing call a repair service.
(Edit)
Oops I see you know how to check the capacitor. To answer your question the capacitor serves two functions, one is for the fan unit and the other for the compressor built into the same unit. (oops, update, *LOL* figures you have two separate ones)
....When you replace it, assuming you know what you are doing, it doesnt hurt to replace the contactor while you are at it.
(For others it's very common for the Capacitor to be a dual function. One part of the capacitor is for the fan motor and the other part for the compressor. So you replace just one part. If you have two like the OP one will be bad and the other good but while you are in there you may as well replace both at the same time and know you will be good for years to come, same goes for the contactor.)