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Have you examined the wire at the outside unit, for both dog chews and weed wacker attacks?
Turn power off to both inside and outside units, Placing a small fuse in the low voltage circuit will help preserve xformers. Was the t-stat in cooling or heating? With your Ohm meter start checking low voltage coils til you find the coil that is short to ground. I would check the outdoor contactor coil 1st. Best Wishes J.Lockard