I vote that you find a good hickory handle somewhere (not lowe's or home depot), and learn how to properly set it yourself. There's really nothing like a properly set handle. ...the tool comes alive in your hands. Don't be afraid, if you're seeking tradition, do what your forebears would have done.
Well said. Take ownership of the axe and fix it. The pride and satisfaction are well worth it, will make it less chinese. A wood handle is much nicer to use than plastic/fiberglass/steel. If you want a better one, that's fine, but fix it first then give it away.
Here's how to set an axe head permanently: remove wedges and head. Glue the splintered piece back on with epoxy or gorilla glue, they both are water proof. Use drawknife or whatever carving implement you have, such as a large, sharp pocket knife, or heavy kitchen knife, and shave handle tenon undersize by about 1/8" with respect to the axe head hole.
Try to match the taper (if any) on the lower half of the drifted hole in the head. Use chalk or pencil marks to help you find the high spots. But remove more wood than you think, we want a loose fit for installing a rubber liner around the wood handle tennon. Shave the shoulder down so that 1/8 to 3/16" of tennon protrudes from axe head when seated.
Then wrap the wood handle tennon with either 1) inner tube rubber, or 2) coat with Sikaflex polyurethane construction adhesive (from HD), and set the head firmly on the shoulder. With Sikaflex, set axe somewhere warm and make sure handle is square to head. Wrap with saran wrap to keep sikaflex from flowing out. Give it a couple days to fully cure. Or a week. Sand/shave off excess Sikaflx, then drive 1 or 2 steel wedges into tennon, and grind off any excess wedge metal.
With innertube rubber, wrap at least 1 revolution, and up to 2 revs, then secure end of ruber with short piece of ductape or electrical tape. Carefully work axe head down over the rubber. This may take a couple of tries to get right. Cut rubber oversize, then trim off excess when the head is seated. Seat head firmly on the shoulder, then drive 1 or 2 steel wedges into the tennon.
The rubber between wood and steel will allow the wood to expand/contract with seasonal humidity changes without crushing wood cells. With properly driven wedges, the head will stay tight for many years, 10 or 20. If a wedge vibrates out, pull it out, put some glue in the hole, then reset the wedge. And the rubber absorbs shock, so its easier on the hands and the handle.