Measure both AC and DC current through the secondary of the
transformer.
If putting a multimeter into DC current mode gets a reading
around or over
50 milliamps (,05 amp), then something is wrong with the load.
If DC secondary current is near or over 200 milliamps, then,
"Houston, we have a problem"!

I have been (sort of) following this conversation. I assume this
transformer also drives the contactor for the AC compressor.
Maybe there is a short in that circuit and the transformer
secondary gets shorted when the thermostat calls for cooling.

David