97 F-150 Neutral Safety Switch?

I have a 97 F-150 2wd with V8 and automatic trans. I bought the truck used, so here's all I know.

I will try to start and get nothing but the dash "idiot" lights come on. No clicking like a solenoid or starter issue. The battery is good and the battery cables are good, tight and clean. When I run into this "no start" condition, I can pull up (using left hand to pull up toward the park position) on the gear shift and the truck will start.

The extra twist. This does not always work. Sometimes, I have to pull the gear shift back sometimes push it forward. Sometimes, none of this works so I set and count to ten, about twenty times and don't have to pull push or move anything and the thing starts.

I'm one of those (I use to be a mechanic) guys but that was back in the eighties. I was thinking this was the Neutral Safety Switch but, I know there's a lot of new buzzers and bells on these newer cars and was hoping for a second opinion.

I was also needing to know if this does sound like the problem, where is the switch and is it adjustable?

Thanks!