HVAC control transformers are notorious for over voltage when lightly loaded. We used to place an MOV (metal oxide varistor,) rated at 30 volts on our electronic thermostats for protection. These thermostats were designed almost twenty years ago. Place a heatsink on your regulator, place the regulator near the top of your circuit board, and place your temperature sensor near the bottom, with the board mounted vertically the heat rising from your regulator will pull fresh room air over the temperature sensor.

Melanie's idea about centertapped transformers would make your project easier, but most residential HVAC equipment sold in the US won't have a center tap. However, you can buy one and replace the transformer in your air handler and take 12 volts off the center tap. The newer switcher regulators don't produce the heat that an LM7805 will, making them a better choice, I wish we had them all those years ago.

Jerry.