When I turn off my AC breaker off in the garage, I expect my thermostat to power off. Initially, it did. Now, it stays lit and I can change the settings. That’s odd. I have even let the breaker off for hours. Still no luck.
How do I turn it off completely?
Many AC units have more than one transformer. There must be another transformer inside the unit just for the thermostat and condenser unit relay/contactor. You should have a disconnect switch at the outdoor condensing unit - try switching that off as well.
You should be safe and use a voltmeter to check some voltage points, referenced to a ground, to be sure all power is off. This includes all 120V or above, and all 24VAC voltages.
The only other possibility is that the t’stat wiring is run in parallel with a different 120V or above power cable. It is not uncommon for the other 120V cable to induce as much as 40V onto an adjacent wire/cable. That is enough, in some instances, to make modern day low voltage devices operate. (Seen this many times in industrial settings.) So, in that case, you would need to identify that other 120V cable and turn its breaker off as well.
Your thermostat is probably running off the transformer in your air handler or furnace, not your air conditioner condenser/compressor unit.