The fire axe is considered the superior melee weapon, after all, it serves three purposes that are all vital for you.
A fire axe is a good weapon as it kills in 1-2 hits, but its downside is its range, which isn't very good, and missing a ghoul will allow it to come dangerously close to you, as long as you don't miss a second time, you will be safe. A fire axe should be your primary weapon. It has average durability.
The fire axe is the superior door-breaking weapon, as stupid as it sounds, it chops doors MUCH faster than other weapons. Other weapons take a considerable amount of time to take down doors.
Probably the most important of the three is the ability to chop trees. If your base is near an area with tons of trees, you can consider wood as a "renewable" resource, because you can start making your own wood. Of course, you will need the saw to process the wood though.
The axe is one of the rarest weapons, but a day or two of thorough searching may result in your first axe, so keep searching.