tom,
The "drill now" point of view would, in select areas, remove pre-existing regulations that prevent drilling, i.e. less government. The oil companies have wanted to drill for years. It shouldn't be left up to a pure race, though, the dirtier fuels win easy. Coal is cheapest, oil next, gas next. There should be help for renewables, and there is.
The 10 year figure is debatable, but its probably in the ballpark. Red tape, rigging, drilling, recovery, etc. all take time. Then refinement isn't quick either. The phrase "till we saw a drop" is not literal, it means until it becomes gasoline and is sold to consumers.
And, your numbers are correct (or at least close) for "technically recoverable oil", which is itself a moving target, there's 3 times that much oil there and more will become recoverable as drilling technology continues to advance. So, if ANWR were to supply 100% of the U.S. oil, it'd last us at least 12 years, maybe as many as 32 years. But thats not how it works, the same numbers say ANWR could supply 50% of our oil for 24 to 64 years, 25% for 48 to 128 years, see where we're going? With the rate you can pull the stuff out of the ground and transport it, we're probably talking more like 10%. Which, based on you're figures, means we can pretty much stop buying oil from Venezuela for the next 120 to 320 years. So the real question is whether to drill in an arctic tundra with a human population of nearly zero and supporting only a small number of animals (yes, there's caribou), or in a rainforest with a high human population and giving the money to a dictator who hates us? By the way, ANWR is only part of the "drill now" plan.
As far as "then what", the running out of oil is far into the future, in other words, if we move away from oil, we won't run out. Nobody, not even the "drill now" crowd, is arguing that we shouldn't slow down our use of oil in favor of alternatives. Its just important to realize that abandoning oil is a slow process. We should use less, drill more, meaning we import a lot less.
BTW, I do know about the wind farm in Delaware and support it wholeheartedly. The great plains area is particularly ripe for wind power.