Neutral Safety Switch

What year did you buy the new switch for? If I'm not mistaken you need to buy it for a 69 or 70 model because later on (not sure exactly what year) they changed the length of it.

And lets address the wiring.

The middle pin on the newer switch is for starting. The outer pins are for the reverse lights and that's why it would only start in reverse the way you had it wired up. All you have to do is run the wire (brown wire if I remember right) coming from the starter relay to the center pin on the newer switch.