Air Conditioning - push started fan but still blowing hot air

My outside fan went out in the night, when I woke up I realized the fan wasn't blowing so checked it and the fan housing was hot and the unit was making a humming sound, which to me ment it was getting power, after switching everything off for a while I turned it all back on and same problem. So i push the fan to see if it was the bearings and it spun fine and then while spinning it turned on. I thought maybe I would need to replace the capacitor, but after running for a while it only got warmer in my house and didn't blow out cool air just warm.
Also I checked all of the connections inside the fan unit and they were good as far as I could tell.
Can a bad capacitor cause the air to blow warm?
I also checked the copper inside the fan unit and it was normal temp

Yes...the same cap can be used to start the compressor and the fan. The fan started because you helped it, but you can't help the compressor. Try a new cap, cheap and easy to replace.

Don't keep trying to run it...you can cause expensive damage.

Thanks gunguy I appreciate the quick reply and the quality info. Do you know the best way to figure out what model cap to replace it with, or should I just take in my bad one for a comparison?

Take the bad one....
And read the thread here..http://forum.doityourself.com/electrical-c-d-c/393703-discharging-start-capacitor-before-removal.html

Make a diagram of the wires before removing anything.

Thanks again Gunguy, I'm on it!

"To replace the capacitor- turn off the power to the unit. Make a good sketch of where the wires go on the capacitor and take a digital photo as well if you have a camera. Discharge the capacitor by placing an old screwdriver across the terminals, remove the wires and remove the capacitor. It should have some numbers on it. You're looking for the uf or mfd rating [uf and mfd mean the same thing, just 2 different ways of saying microfarads] and the vac rating. For example, it might say 30/5 uf 370 vac. On the new capacitor you want the same uf rating/s. For vac you can go up but not down. Example- if your cap is 30/5 uf 370 vac, the new one can be 30/5 uf 370 or 440 vac, but if your cap is 30/5 uf 440 vac, the new one has to be 30/5 uf 440 vac not 370 vac. Motor runs caps on a/c units only come in 2 vac flavors- 370 and 440. So, install the new cap and monitor unit for operation."