I've been shower-shaving with my Wacker 15/16 all summer and have had no issues with rust or staining, even in the pivot area.
Last time I visited LA, I was under the impression that it rained 3-4 days out of the year. It may be okay to take a shower [while] shaving with your razor there (although I wonder about the water consumption involved), but in more humid environments, leaving water in the pivot area can be a real issue.
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