creating a circuit using half/full adder
Using an adder seems like an odd way to solve the problem but I think the teacher is looking for something along the lines of this:

Use a 1 bit full adder with it's output starting at 0.

Car red and people green = adder output.
Car green and people red = not adder output.

The 3 inputs to the adder (a,b and carry) are the two buttons and the current output of the adder.

So when a button is pressed the adder output increases (or if it's 1 wraps around to 0) and the lights change.

You would need to include a latch or play games with clocks to prevent the system counting as fast as possible while a button is held down.

It all seems a little overly complex but I can't see how else you would do it using an adder. A latch and a couple of logic gates would be a lot easier.
I am making the assumption that the buttons are momentary. i.e. press and release the button and the lights change state and remain in the new state.