You only need one adapter, and that's for the fan, providing that it's a 12v fan. You need 120volt power from your house plugs to operate the light. If you use an extension cord, cut one of the wires, and hook it up like you have it in your drawing above on the thermostat and hook both wires to your light.
You will have the red and black in your drawing going from the adapter to the fan. Then you will have the black and white from your house power going to the thermostat and the light