Reducing dependence on imported oil might not have the desired effect of reducing gasoline prices. Just saying.
The oil price now is function of 3 factors. Normal supply/demand force that is vectoring toward ever more increasing demand by developing China and India and is here with us to stay unless China and India will get into a hole. Second is market fear. Fear of losing supply because of Middle East instabilities. Third is play by speculators that hedge that the future prices would go up and that its a good place to invest money. Dynamic that is very similar to all previous bubbles, IT, housing.
So, while drilling is not going to reduce demand and drastically increase supply, it will reassure markets that instability risk can be met. Add to that removing barriers in nuclear power development (that when is combined with emerging pluggable car technology can indeed reduce oil consumption).