Solution: a bathtub. I have some combination bathtub + shower so I sit down and wash myself -- it doesn't take longer and I can fall (since I am already at floor height.
I can easily think that climbing in and out of a bathtub is way more risky (wrt. falling) than merely stepping under a shower. I believe people don't fall when they stand in the shower. They fall when they step out of it on moist surface. Sorry if I ruined your day. :)
I visited a friend in Slovenia and discovered this is the normal routine in Central Europe. In fact they have comfortable tubs which have a higher section for sitting on.
