gabel has it... dry wood. I use ash as that's what my area does best.
take the axe head to the handle, figure out which end is up and down, measure for any difference in the size of the hole.
place the hole on top of the handle and reach inside and trace the hole onto the handle. take a rip saw and cut down just outside the line(this is the time to shoot for the other end of the handle). then take a drawknife and get closer to the lines. finally bring the top(the very end-1/4") of the handle just under the line, and begin fitting the head. rub some lumber crayon inside the axe head to show the tight spots quickly. there is no substitute for patience at this point. work the head on(I use a file here), keeping the bit pointed toward the other end of the handle. A scoring axe, when set on a flat surface with the bit pointing straight down and the end of the handle touching the same flat surface, should have the center of the bit on the surface. THis means that the section of handle that passes through the head is tipped a bit in relation to the rest of the handle. Then when you swing the axe, the center of the bit is hitting the wood first.
The adze is the same, in reverse. the whole handle goes through the head, and then gets fitted.... but you can still trace the profile that you need to end with onto the end of the handle.