When it comes to fossil fuels, there's no such thing as 'energy independence' because fossil fuels are traded on a global market. In addition, not all oil is the same. So you have to trade it because what you can extract may not be what is useful to you, but useful to someone else.
In short, the US cannot functionally be independent on fossil fuels even if we extracted every drop of oil within our borders--because we literally cannot use all of it, and most of it would be wasted just sitting around.
In extreme scenarios it is certainly possible for the US to ban exports of certain fossil fuels, effectively making an internal US market that is isolated from the rest of the world and basically energy independent.
Yes, we would need to build more refining capacity to use it all effectively, but in a cataclysmic event (i.e. a world war or something) the US would be much much better off than most other countries.