I have an older York unit. The compressor began to have trouble starting. The lights would dim momentarily and the condenser fan would run, but the compressor would cycle on the overload klixon. After a few tries, it would start.
As a guess, I ordered a new run capacitor on ebay. In my case it was 35uF 440V
When the cap arrived, before replacing it, I used a clamp on amp meter (if you are going to do DIY HVAC, you should buy one of these) and checked the current in the run winding and the start winding while the compressor was running.
The run winding was 20 Amps and the start winding was nearly 0 Amps.
Then I replaced the cap.
The compressor started much faster and the new readings were run winding = 12.5 Amps and the start winding 6 Amps.
So the lesson is, if you suspect the run cap. use a clamp on amp meter and check the current in the two windings. For a PSC motor which most domestic AC units use, the start winding current should be a substantial fraction of the run winding. m