Why does home air conditioner stops blowing when the unit is working and putting out cold air?

There is usually a diode or resistor in-line with the power feed to the blowermotor in your A/C unit. If the blower works then cuts out on you it sounds like your diode or resistor is going to burn out soon.

Take the A/C unit to a reputable tech, it's probably just a quick blowermotor swap out and you'll be back in business.