Should I wear my oxygen in the shower?

If you need to, sure. If you cannot go without your oxygen support for more than 30 minutes without having breathing or related medical issues, then yes, get a long enough oxygen tube, leave the tank outside the shower, and have at it. No problem.
It depends. It depends on the mode of administration of oxygen - whether you're using nasal cannula or face mask. With nasal cannula you can, but it may not be feasible with the face mask. It also depends on how long you can tolerate being off supplemental oxygen (depends on how long you shower!) & what the underlying cause/disease process is. Oxygen should be handled carefully because of the fire hazard.