I think I would use ash for the stick. It's hard and just a little springy. Hickory, if you can find it, would be even better, and you could use it for the entire cane.
>I know if the grain goes up an down the whole way the handle could easily snap as soon as it goes past the edge of the support.
Attach the handle with a tight mortise and tenon joint -- just like hammer heads are attached to their handles. Don't peg it on, as that would weaken the end of the stick. This is the old way to make crutches, by the way. It's strong.
Me, if I were to make one out of a single piece of wood, I'd look for a branch that grew nearly straight out from the tree. The branch would make the stick, and a section of the tree where it grew out would make the handle.