The metal flu that you are describing is called B-Vent and it should be ok for burning wood. Make sure it is of proper diameter for your wood burner. I think your money would be better spent on replacement windows and at least R-30 insulation. Then look at your heating equipment, maybe it is time to upgrade to a 90+ efficient furnace.
I have a 3000 square foot house in NE Ohio with new Anderson windows and a 94% Rheem furnace and decent insulation. My gas bill for this month was $147, that was for an actual reading and my hot water, stove, clothes dryer and built in outside Grill(that is used almost every day) are all gas. That bill was also for a month that had a high temp of 21 degrees. Most nights were near zero. I have some friends whose bills were very high, they all have very little insulation and old furnaces.
I only have R-19 in the attic, this year I will double that and expect to use less gas next year.
I think your money would be spent more wisely by insulating and upgrading.