that's a really fun job and a pleasant way to spend an afternoon. it would really help to have a model to go by. usually this is the old axe handle from the same head.

here's my process:

working from a clear log (i use black ash because that's what we have) a bit longer than your desired handle

split it in half, starting with an axe driven by a sledge, then a maul in the crack opened up

then you get to see the grain

quarter the halves

maybe one of the quarters has a nice curve to the grain that can be incorporated into the handle- pick that one

split the quarter to a 2" board

even it out with a sharp axe

lay out the axe handle shape on the board

rough it out with the axe

smooth it with a drawknife

take it from square to octagonal with the drawknife

then round it out nice with a spokeshave

maybe a rasp for the knob if you like it round, or just saw across the swell at the end of the handle. both look nice imho

spokeshave the other end of the handle to fit in the eye of the axe

saw a cut in the end of the handle for a wedge

bang the head onto the handle using the weight of the head

put a hardwood wedge in as far as it will go

put a metal wedge in diagonalwise

oil the handle

start a fire with the trimmings

i think a split handle that follows the grain is far preferable to a cut one- stronger. the handles on most store-bought axes, even decent brands like an oxehead, are pretty poor. this sounds like lamenting the good old days- and i wasn't even there!