I would go after lumber and pre-dried material to keep "wet" wood burning. There is always some new construction somewhere near by and if I was desperate enough, I would leave a note saying, "I took this due to the emergency, Please call me to settle the debit"

The issue of how to cut it, is answered by the three axes that I keep in my truck at all times (which is odd because I am a computer geek) I also keep a small folding bow saw. If the wood was too big for that, then I can use one of the Axe's that I have.

keeping all of this in mind there is always a way to turn a table saw into a foot saw that could cut wood. (attach a large spinning wheel and a peddle and you are good. for that matter why not retro fit a chain saw blade to a bicycle to cut wood if you think that the issue is going to be that long.

while I am very proud of my ax collection, I have to add that once a log is chopped you have to split it and the ax leave a very pointed end so you cant really easily split the wood thus making it harder to burn the wood. So you are rather stuck with smaller pieces that can be cut with bow saw.

Having actually heated a cabin with a bow saw and ax I can tell you that it is no small feat. So I would again go back to a bow saw and nice kiln died wood that you find at construction sites. I don't condone stealing and advocate that you try to find lumber by other means, but if push comes to shove I would be going to the lumber pile for my supply of wood.