When the temperature reaches the desired setting, the gas burners turn off as they should, however the fan continues to run and blows cold air. The temperature dips well below the thermostat setting, but the burners do not turn back on and the fan continues to run. I also notice a flashing red light. I can only get the burners to come back on by turning off the power to the furnace and turning it back on.

The flashing light is a fault code , look for a label to explain the codes and take the corrective action.

Power interruption reboots the logic chip on the board.