Showers are fine. It really doesn't even shower in Southcentral--the rain there is more like a falling mist than anything that actually gets you wet. No one I know in Anchorage ever uses an umbrella (I lived there for 17 years and didn't even ever own one).

In Southeast, it actually does rain (real rain), but they don't use umbrellas there as a mark of pride (kind of like how Fairbanksans will go out in shorts at -40, just to prove to outsiders that they're tough).