Why spoil the ship for a haporth of tar? I'd say strip the old one out, bin it, and do a new one. I personally do not rate store bought axe handles (or adze or hammer handles) as they are almost always carelessly produceced from sawn stock, which allows weak spots caused by cross grain, and in any case they still often need extra work with a file etc to acurrately fit it up to the axe head. I prefer my own air dried cleft ash handles. They might look slightly more primitive, and slightly "wonky"
but they are far stronger. And if you do remove the handle dont throw the axe in the fire to burn the wood out in a hollywood backwood's fashion, it will only ruin the axe temper. I use an old chisel which I keep specially for the purpose to chip the remains out bit by bit. The metal wedges are usually re usable