Pushing it with a stick doesn't fix anything. It's just a test to help diagnose. The bulging is not necessary for the capacitor to be bad but if it is bulging, then it's bad. I'm pretty confident it's your problem but obviously can't reassure you that without being there. For the cost, it's worth a try, and if that's not it, it will likely be one day, so you at least have a spare. Lastly, you won't get cold air just because the fan spins. The compressor is not working still and again; the "stick test" just helps you determine that a capacitor is a likely suspect. Shut it off and find your replacement part. If it still doesn't work, then I would call a technician.