Does Edward Cullen shower

Most likely not, because Bella had said that she wanted to be human for a moment and went to take a shower. This probably means that vampires do not take showers.

OR...

Yes it is true that Bella said that she wanted to be human for a moment and take a shower, but she then said want to come in and help me wash the feathers off? When he said he had to make breakfast that implied that he could take a shower but chose not to out of guilt of bruising Bella.