I'm really glad I recently replaced my old 80% efficient oil boiler with a 97% efficient natural gas boiler. My winter bills are less than half what they were and it barely uses any gas during the summer for hot water.
With 80% efficiency, you have to burn 1/0.8=1.25 units of fuel to get one fuel unit worth of useful heat. With 97% efficiency it's 1/.97≈1.03 units.
So if fuel prices and the amount of useful heat usage stayed the same, then your bill should be 1.03/1.25≈82% of what it was before. If your bill is "less than half" then something else must have changed.
Yes, I switched from an oil burner (#2 heating oil) to a high efficiency natural gas boiler. The other key ingredient is that the oil boiler was 20 years old and did not have "cold start" (so it always keeps itself at temp even in the summer).
Any modern boiler (gas or oil) will almost certainly be cold start, so it only fires when there's a call for heat. This alone probably saves a gallon of oil a day at least.
There's a reason the Energy Saving Trust over here in the UK discourages people from replacing their gas boilers with heat pumps thinking it'll save them money, even though our climate is more suited to it than the colder parts of North America. Unfortunately, that doesn't stop our press pushing the idea we should go for heat pumps because Germany has. (They don't have such widespread natural gas connections to homes as the UK and so are reliant on oil-fired home heating, which as you can see has very different economics.)
We also don't know what the long-term fallout is going to be. Natural gas has been unusually cheap recently because it's a byproduct of fracking for oil. But low demand for oil means less demand for fracking which mean less natural gas production.
It's really hard to predict where prices are going to be in a year, because you really have to predict how well the oil industry can predict future demand in this environment. If they shut down too much production and then demand comes back there could be a price spike. If they expect a quick recovery and are wrong, prices could remain on the floor for a good while.