I't probably a matter of some minor required maintenance. If the car has beeen very reliable for a long time, you may have taken it for granted, and perhaps neglected some adjustments that are typically necessary.

I would not suspect the neutral safety switch as a culprit. If that fails, it may prevent starting, but not likely make it stall.

First, I would check your ignition points, distributor cap and rotor and spark plugs, in that order. One good indicator is to remove a spark plug, plug it back into the wire, lay the body of the plug on the valave cover so it makes ground, and look for a bright blue spark when you crank the engine over. Pale or weak spark may not be enough to keep the engine idling properly when shifting into gear. If pale or weak, procedd to the distirbutor for an examination of the condiditon of the cap and rotor and points.

Most older car repair manuals can guide you through cleaning and adjusting points. Get yourself a Haynes manual ideally.

Prior to 1975 all these cars had ignitions systems that should get periodic maintenance. Other than that, the fuel filter should be replaced once in a while, and of course the oil and filter and air filter too.