Sounds like your neutral switch maybe out of adjustment.
If you put your head in the floor board under the steering wheel just below the dash you will see the neutral safety switch on the steering column. Loosen the screws and move it one way try it. If that don't work loosen it and move it the other way. HAVE YOUR FOOT ON THE BRAKE WHEN TRYING TO START HER! Adjust till you get it right. It should not start in gear!
Richard D. Hord
'60 Thunderbird "Christine"