I have been dependent on a cane to get around, I am thankful that I currently have myself into a type of condition where it is not needed, I get around just fine and can walk for miles. I did put out the money for a blackthorn walking stick years ago because I had inherited a proper Shillelagh many years ago and have been very impressed with its unbreakable strength when needed. It is however too short for my continued use as a walking stick.
On the matter of using hardwood dowels for a cane I will mention a word of caution. Being involved with black powder firearms I have needed to build a ram rod or two for many rifles and shotguns, it is the nature of the beast...we do after all load our rifles and shotguns from the wrong end and need something to cram everything down the barrel. The only readily available hardwood that is recommended for ram rods is hickory, oak has a nasty tendency to split into two very sharp pieces that the user has a very good chance of running one completely through his hand, I have seen it and it ain't pretty. If your dealing with oak over 3/4" thick it is pretty tough to get that to split but at the diameter of a loading rod of 3/8" or so it can happen. Hickory is the only wood recommended for a loading rod that is readily available and can be purchased from a variety of outlets.
That being said your laminated hockey stick is more resistance to splintering than any known wood, I would just be careful substituting oak unless it is thick enough. I have a boe that I got while practicing martial arts back in the 90's, my Sensei gave it to me upon reaching purple belt. It is probably red oak and has some of the tightest grain I've seen, it has zero flex as far as any human being's strength goes, I have done pull ups with it, I weighed around 230lbs at the time. I carry the boe when I am hiking off road or huckleberry picking, very dependable.