How to wash her hair when she can no longer handle the shower?

My mom is so scared of getting in the shower. She will do anything to get out of a shower, including lie and say she just took one. (Lol). She whimpers and says she feels shaky. It makes her nervous.

She only showers once a week and that is only so she can wash her hair. Every day she gives herself a "whore's bath" (her words). In other words, she washes thoroughly in the bathroom sink with a washcloth. She washes all parts.

She is mobile. She can walk and make her own breakfast and lunch. She's got some dementia (undiagnosed but it's general - short term memory, some confabulation). She DOES get confused about which way is hot, which way is cold and which bottle is the shampoo bottle.

I sit in there on the toilet the whole time. I hand her the shampoo and remind her which way the hot and cold is. I hand her her towel. She has the sink and bars she holds onto to get in. It's a regular bath and we are renting in independent living. (I live with her - that's why we don't need assisted living which would be unaffordable.) Oh and she can't stand strangers. She's a highly nervous person and has even developed this nervous vocal tick (which upsets me.)

It takes an hour or so to get her shower-ready; the bathroom warm enuf with the heater and talk her into going in. If I could wash her hair in the sink I would but it's physically not possible. The sinks are too high. I got a shower extension and a seat and tried just having her sit in the seat. She yowled and howled for that too. Poor thing. It stresses her and stresses me. The only alternative that I can think of is taking her to the beauty parlor once a week to get it shampooed and set (I set it). But leaving the house freaks her out too. Not as much but she says she doesn't want to do that.

I dread Friday - Shower Day. So does she. Any suggestions appreciated. Btw, she's 103 but looks 89.