Help diagnose a compressor problem
Have you checked the actual voltages feeding the fan motor ? 


You should use a volt meter , measure both at the contactor(both in &amp; out sides) and the motor. I would like to know L1 to Neutral , L2 to Neutral and L1 to L2 as there is a slight possibility your actual Circuit Breaker is "broken" on one leg where you might get a reduced connection and thus not enough current passing.
Have you cycled your Circuit Breaker Off/On in your service panel ?


Your breaker may have partial tripped, only half the breaker tripped and so it has partial connection (brown out voltage basically).


Also when you replaced the capacitor was it a dual fan/compressor
type ? 
Or Start/Run type ?


What this looks like is that your fan has a start capacitor and that gets it going but your run capacitor is not working.

Please post your Model number, I know it is a carrier - inside the panel there should also be a wiring diagram - post a good picture of that if you will.