neutral safety switch

Good diagnostic work. Yes, you need a Neutral Safety Switch. You don't mention what year and model your car is. The switch is screwed into the driver's side of the transmission, just above the trans pan, near the front. It's different for '60-'64, '65-'68, and '69-up. Only the latter ('69-up) switch is easily available locally; the others are harder to scrounge up. No repair is possible, service is by replacement.