The neutral switch will keep the starter from engaging while in gear. It should have no bearing whatsoever on whether it stays running after it is started.

Just so we're clear, shifter in P or N, no start as in the starter does not engage?

Is the neutral switch on the trans or in the shifter on those? IIRC some console shift cars had a switch on the shifter but I forget if it was neutral and backup lights or just backup lights.

If the trans responds to the positions on the shifter ie shifter is in Park, trans is in Park, shifter Reverse, trans Reverse etc, then I don't think your cables have an issue. They really have no reason to "go out of adjustment" if they were right to begin with.

Kevin