So what do you think is going on now (as described by the Chron)?
I don't think anyone really knows what is going to happen next in the economy (as in so many other things!). Sometimes people focus on changing just one variable at a time in order to simplify the analysis. So an increase in oil prices could lead to an economic slowdown. And an increase in oil prices could bubble through the economy and cause an increase in many other prices. Put these two pieces together and you have stagflation.

But they didn't do the analysis all together. If they did, they would see that the economic slowdown will reduce demand and curtail the amount people are willing to spend, making it difficult or impossible to raise prices. What might end up happening is that some prices will fall, those for which the effects of decreased demand are stronger than the effects of increased costs.

I am not an economist but what I understand is that the stagflation of the Jimmy Carter era is seen as due to bad policies on the part of the central bank. Basically they had this Keynesian model where the bank could choose any point they wanted on a curve trading off unemployment and inflation. But after playing this game for long enough, people began to factor in future inflation, building it into their contracts and such, and as a result inflation no longer had the stimulative effect that it used to. Basically Keynesianism only worked if nobody knew about it. Once people caught on to the con, the jig was up. That bill came due in the stagflation era.

I do agree the central bank has a role to play - if they raise interest rates quickly enough, they'll be able to head off inflation and prevent it getting culturally established. I lived through this particular set of policy choices in the UK in 1979 (Thatcherism) during the worst of the oil shocks. It did indeed cure inflation, but the interest rates required to do this were extremely high, and the economy became very distressed in other ways (massive unemployment, large scale loss of industrial base). OTOH, if one keeps interest rates low through an energy crisis, there'll be galloping inflation as the happily growing money supply chases less physical economy.
I should add - I strongly assume that the Greenspan fed will choose inflation fighting first, if it has to choose.
But, the main question isn't that rising cost will destroy demand.  We know that, at some point, that is true.  It's just a matter of determining how inelastic the price is.  Is it $3? $5? $10? a gallon?  Personally, while I feel the pinch at $3, I haven't changed many habits besides for cutting down on maybe one extra trip a week.  I'm certaintly not at the point of buying a bicycle.  Then again, my daily commute is about 2 miles one way.  Being unspecialized labor, I'm lucky in that area.  

As far as the relation between unemployment and inflation.. I remember the curve you're talking about.. something makes me think it's called the Rule of Seven, where for every point of unemployment gains, you gained 7 points of inflation, although that obviously can't be it.

If there's any economics specialists around..