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I may have made this up though
To a large extent, you have. While commodity consumers will buy futures, the price of the futures won't necessarily be the same as the spot price on the day they buy them, since the person selling the futures will have an opinion about how the price will move in the future. Also, when that futures contract turns into a delivery, you'll want to sell the commodity on at the spot price on the day of delivery, not the price you paid for the futures contract. There's no point selling it cheaper, partly because that would be plain dumb at any level, but also because you need the profit on contracts where you bought the future for less than it turned out to be worth to offset the losses you make when you called it wrong.
The existence of a futures market has a stabilising effect on the spot price over time, but it doesn't in itself cause any price lag on spot trades. Your pump price is set according to a wealth of factors beyond crude prices, but the element which is directly related to crude prices changes on the same day as crude prices change. There is some apparent lagginess caused by other factors, mostly local market competition causing people to hold prices down to avoid losing market share, in a sort of Dutch auction.
I thought that the energy companies bought a supply for a certain period at a certain price, so shocks such as rapid rises in crude/FX issues took a while to come through. I may have made this up though.