While historically there has been a correlation between the price of crude oil and natural gas, that correlation appears to have broken down in recent years, as crude oil has surged and nat gas has done its own thing, mainly due to large fracking supply increases. These days, there is no expectation that because oil is falling, nat gas will fall in tandem, or vice versa.
As you noted, since nat gas is a domestic market, not an international one, the strength of the dollar will not directly impact the price of nat gas the way it does crude oil. Essentially, weather and storage / supply are what set the natural gas price in the near-term. Indirectly, falling crude oil prices might draw in more traders to trade natural gas, as they see it as an opportunity to profit from a seasonal energy trade. But, that is hard to quantify. Keep an eye on storage / supply and weather outlooks. It is important to note that nat gas reacts to future weather, weeks in advance. For example, this week is warming up across the eastern U.S., but nat gas has spiked in price anticipating cool weather predicted for coming weeks.
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