If I do and take some of that, guess I can clear my carbon conscience by not driving. 
This turns out to be a complicated question/calculation. The last I looked into this, my memory (so take with a grain of salt) is that generally, using wood that is harvested and burned is worse than just burning gas to heat with. There's a lot that goes into this, but essentially it boils down to black carbon being really bad over the short run and the "renewable" aspect of trees only plays out in the very long run. Since we need to get GHG output down rapidly in the short run, burning wood should not be considered a renewable fuel at scale.
Also, there are huge differences between old woodstoves and new ones. New ones typically route the exhaust back through the flame and get a secondary combustion that cuts down dramatically on the pollution they put out. There's basically no scenario in which burning wood in an old woodstove is preferable to burning fossil fuels from at GHG perspective.
But your situation is more like ours - wood is dropping to the ground all the time. I just cleared a large tree out of the street last week. Trees threaten houses. We need to clear for fire safety and so on and so on. Most of that wood rots. The question is what is the carbon cycle of rotting wood and what gasses does that emit and how does that compare to burning that wood and pumping out the black carbon?
That seems to be debated by experts, many of whom have vested interests in one argument or the other. But in general, it seems like electricity is usually better unless there's a lot of coal in the mix, natural gas and even oil is better than wood harvested for heating, and it's close to wash but probably slightly better to use "waste" wood for heating than to use gas.
So there's a rough guide to how much conscience needs clearing.
In our case, we do not have wood backup actually, but we definitely cannot depend fully on electric heat because of our numerous power outages, especially in winter. So if we pulled out the gas furnace, we would need to add in a woodstove. As it is now, we can use the battery backup to keep the gas furnace running. Previously, during a 5-day outage in the winter, we had to run a generator to run the gas furnace to get temperature up and then let it fall through the day. That was probably almost as bad as burning coal! But hey, the roads were closed, so nobody could drive :-)
