I always find things in the last place I look, too.
My symptoms were similar to yours, but not identical. Mine would not start at all. But that your security light stays on points right to the VATS system. If the contacts in the ignition switch (officially they are called 'leaves') are worn, it may be that they make good contact sometimes and not others. It could be totally random, i.e. not related to the time since the last shutdown.
You will need a wiring diagram; find the two leads coming from the steering column that go to the connector under the dash left of the column. Cut the two leads, and insert a resistor of an appropriate value. I had to combine several in series to get the right value, but that'* not hard. I stripped the leads and soldered the resistor across the two leads, and then covered all of the non-insulated portions with heat shrink tubing.
It has started up flawlessly ever since--and no security light.
I would bet money that this is your problem.