The power you are turning off is the power to the HVAC system that sends power to the thermostat. The batteries in your old thermostat do not matter. If you have a split system the indoor unit, the air handler and possibly gas furnace is the unit to turn off.
Ken, The AC Cooling Wizard
We appreciate your help here, CoolingWizard.
BTN, I wanted to follow up and see if you are still in need of any help. Please let me know if you are still having any concerns or questions from here, as I would be happy to take a closer look and assist you further.