Problem with a custom input handler

I am not satisfied with the built-in "Input.GetAxisRaw" method for controlling a character. I want a value of "0" to be returned ONLY when no keys are held down, since this method still returns a value of 0 if both keys on the same axis are held down (ie. left and right) Here is my attempt:

         //if right is pushed first
         if (Input.GetKey("right") && !Input.GetKey("left")) {
             rightOn = true;
             leftOn = false;
         }
         //if left is pushed first
         if (Input.GetKey("left") && !Input.GetKey("right")) {
             leftOn = true;
             rightOn = false;
         }
         //reset if no longer pressing
         if (!Input.GetKey("left") && !Input.GetKey("right")) {
             leftOn = false;
             rightOn = false;
             move = 0;
         }
         
         if(leftOn) {
             if(Input.GetKey("right")) //doesn't register here
                 move = 1;
             else
                 move = -1;
         }
         
         if(rightOn) {
             if(Input.GetKey("left")) //doesn't register here
                 move = -1;
             else
                 move = 1;
         }

I know it seems a bit hack-ish, but it mostly works. What ISN'T happening is that it doesn't register the areas that I've marked with the obviously labeled comments. When I hold down a key, I move in the intended direction, but pressing the opposite key while holding the other key still allows it to continue in the same direction instead of switching directions.

If you know of a more efficient way to get this same effect, please feel free to suggest one.