By you have made 3 axe handles, do you mean you have hewn extensively and thus worn out 2 handles?
What is your experience with the use of a roughing out axe? It is a finicky tool,, but when you figure it out you can get some incredible results out of it. For most timbers, I never even bother with using a finishing axe. I can get more than satisfactory results out of this tool.
My narrow-bitted Bundaxt has a bit maybe about 2 1/2 inches wide, very narrow. Again, it is picky but once you learn it it will work wonderfully. I use it to chop very narrow notches (less work) which it can do since it is so narrow, then split the waste from the top. Using this tool, you can actually get a really nice finish. The rough hewing axe is actually unnecessary, but is a big time saver. A common method is to trim the faces down to their final dimensions with the narrow axe, and if a smooth finish is required use the goosewing to remove a paper-thin layer of wood, along with any irregularities. I have done this, it is actually quite easy, but takes a good while. If you have the roughing axe, you can trim the waste to about 1/4" from the line, score the face and use the roughing axe to trim it down to final dimension. My preferred method is to stand on top of the log and let the axe's weight do all the work, though if you want to move faster you can stand beside and chop down, which is tiring. You will have to chop down around knots, or the grain will cause a great deal of tearout.
And by the way, this 3 axe method inherently results in a sharp bottom corner.