Ignition

Start by measuring the battery's voltage. It will be 12.6 volts if it is good and fully charged. If you find it is around 12.2 volts, it is good but discharged. In that case, charge it at a slow rate for an hour, then measure the voltage again with the engine running. It must be between 13.75 and 14.75 volts. If it is low, suspect the alternator.

If the battery measures around 11.0 volts, it has a shorted cell and must be replaced.

The starter relay is run by the engine computer, not the ignition switch directly. A discharged battery will cause the symptoms you described. Computers do weird things when the supply voltage is low.

To add to the aggravation, the neutral safety switch is no longer simply in series with the starter relay, which has worked fine for over fifty years. It is an input to the engine computer too.

When I worked at a very nice family-owned Chrysler dealership, I had a no-crank condition caused by a defective engine computer. If that is what is causing your problem, you can bypass the starter relay to crank the engine, and it will run fine. I do not remember which type relays you have. My drawings for relay terminals are on a different computer that crashed yet again, so I cannot post it right now. Instead, another way to bypass it is to remove it, pop its cover off, reinstall it that way, then squeeze the contact. The engine will start if the ignition switch is on.