Need some home A/C help/advice/thoughts

Your shop will more then likely have a motor for you, it will be a universal motor that will hook up to most any applications, Do what momoparman said and be sure to shut the main power down.
This does sound like the starter capacitor to me, because if the motor was bad it would just hum and feel locked up. But!! could this be a starter relay sending single leg to the motor ?? Not sure, Is this a heat pump system ?
If your coils are froze up I would call home and have some one turn the a/c off and turn the fan on continuos run, there could be a safety switch letting you know your evaporator coils are froze up and and needs to be cleaned or the fan blades on your air handler is built up with dust and it will not move enough air and have little to no air flowing threw it.
I would turn the fan to the run selection in your house and have someone feel how much if any air flow you have. By running just the fan it will slowly melt the block of ice you may have.