Blocked is Bowser:

So, the Saudi's sell the US oil.

Saudi sell about 1/2 of their oil export to far east. And about 1/5 of their oil export goes to US:

http://www.eia.doe.gov/emeu/cabs/Saudi_Arabia/OilExports.html

The US imports most of it's oil from Canada and Mexico- Canada is biggest oil export to the US. The US buys roughly same amount from from Saudi Arabia, Venezuela and Nigeria- though this can vary monthly/yearly:
DOE

The only solution is for the US to drill in Alaska so that we can burn that into greenhouse gases and a net nothing, but have a lot more dead and starving caribou.

Drilling oil in Alaska might get as much oil as we import from Saudi Arabia, but it wouldn't mean we would stop importing oil from Saudi Arabia. We would import less oil and generally import less from Saudi Arabia. Because the US imports less oil, there probably be less US price fluctuation due to instablity in other parts of the world. The opposite is also probably true- the more we are dependent on imported oil, the more instablity is caused by exterior events. And of course drilling oil in Alaska is not going to kill caribou.

Bottom line: drilling oil in Alaska will not give enough oil so that US will not need to import any oil. Though if more areas in the US were opened up for exploration and drilling, then you could have enough oil so US wouldn't need to import oil. BUT even if the US didn't import any oil, the middle east and other oil exporter, still would a matter of national security interest to the US, and US oil prices would still reflect world oil prices. Now, there is little doubt that using Alaska oil would lower US trade deficit and obviously add to Alaska's economy. And getting to the point of not importing *any* oil, would probably turn our trade deficit into a trade surplus.
Now, in the above, I'm NOT including US shale oil reserve, the known US shale reserves is expensive to extract [though economical at today's oil prices] and could by this reserve alone provide ALL US oil needs for 50-100 years.