So, I ask the energy experts here if this is true. Will the result of more drilling for oil only result in exporting it, and not diminish the U.S.'s dependence on foreign oil imports?

I am no energy expert - but I do know some information that might be helpful.

Yes, the U.S. is a net exporter of petroleum products, and has been for three years now. To the tune of about two million barrels per day, currently. We're one of the world's larger exporters, and the largest producer.

It's also true that there's not an exact match between the oil we produce and the oil we use - both in terms of grade and in terms of geography. We also import a fair amount of oil from other countries (like Mexico and Canada) and then export refined product - essentially "exporting" our surplus refining capacity. Which all means that even though we're a net exporter, we're not entirely self-sufficient. So if we were to both expand our domestic production of oil and refining capacity, it might increase our independence somewhat.

eia.gov: U.S. Net Imports of Crude Oil and Petroleum Products (Thousand Barrels per Day)