There's no way that the neutral safety switch can cause the starter to continue to run after you release the ignition switch.
It only prevents the starter from running when the transmission is not in park.

It is merely a open or closed contact switch that is in series with the ignition switch and the starter solenoid. When it is closed (transmission in park or neutral) power is able to flow from the ignition switch to the starter. When it is open (any other gear selection) the circuit is broken and the starter will not operate.

If the start inhibitor switch could control the starter, you could energize the starter merely be moving the transmission shifter in and out of park position.

Do you have a digital or analog electrical test meter?
Digital meters are notorious for giving a false readings when checking continuity.
You can have much better results using a 12V test light to check for continuity.

It sounds to me like your problem is most likely a defective starter solenoid.

hope this helps
steve