I still have some residual vertigo so standing in the shower and washing my hair and moving my head around can also be problematic. Sometimes I almost drop. Hanging my head upside down in the basin would be impossible. ugh

I would suggest getting a shower chair for safety, even if it's just used for when you wash your hair (or even one of those safety rails screwed into the tiles might help). If you don't have it already, try using a flexible shower head extension. Those things may allow you to keep your head still.

I'd also suggest trying a swimming chamois for squeezing the water out of the hair. They're very highly absorbent and you could squeeze rather than rub your hair and head. They're called different names in different places I think. Chamois, aqua-towel, 'shammy'.

Some days I just dread having to stand in the shower. Doesn't help in mine that the floor doesn't seem solid/stable. I know it is, but it has a little 'give' in it. I always use a rubber mat now so that I feel more stable.

Anyway, good luck with it all and let us know if you find some solutions.