Unless you live in an area of the country where you are heating primarily with soft, straight grained wood like spruce or aspen that splits easily with an axe, get a maul. An axe is going to just bite into the most mixed hardwood rounds and get stuck, especially if there are any knots. Fortunately for your tight budget, maul quality is less important than axe quality. Really, any maul will do. They don't need to be razor sharp, nor can they be with with their blunt angle. When the edge gets too rounded, just grind a new edge. You can go the fiberglass handle route that will be less likely to break than a wood handle, or just go the wooden handle route. You can wrap a wooden handle just below the head with some wet rawhide or paracord to provide a little protection. If it does break, just pick up a new handle and rehaft it. You can probably find a used maul at a flea market for literally a few bucks, alongside of old shovels and garden rakes.