You could ask if the mobile hairdresser will wash it for you, most of them will, I sometimes get them when my anxiety flares up an can't get to the hairdressers, she's really good, gives me the same service as if I was at the hairdresser's an will wash an dry it, even dye it etc, she also lets me know the quiet times at her place too so that when am feeling ok I can go in when there's little to no one there, I know they aren't all the same but mine has like a little bit up the back that's sorta hidden from the front so she takes me to the back an anyone else thats in is down the frontI need to pluck up the courage to go to a hairdresser, my hair is really thick & frizzy although I straighten it, I've probably only ever been to the hairdressers a handful of times in my life, when I was 10 I went she washed, cut, blow dried my hair & I came out looking like Deidre Barlow (had similar glasses too) full on poodle hair, my uncles ex wife was a hairdresser so she mainly cut my hair in my teens, then her & my mum would sit & smoke/drink afterwards taking the piss out of my hair, I then had my hair done in my late 20's at a hairdressers, I spent the 3 hours (wash, cut, colour, blow dry, straighten) apologising for the state of it because its so thick & frizzy, I last went about 3yrs ago, the hairdresser was so lovely but I just felt so out of place, felt like everyone was staring at me & just uncomfortable, then felt they were probably talking about me afterwards about my awful hair, I suppose I could have a mobile hairdresser in but its the faff of washing my hair myself I can't be arsed with (I can't be arsed at the best of times) I just trim my own hair/dye it but want something different. Sorry for the long rant.
Maybe you could ask what the quiet times so that no one is really in since you are nervous about it, most of them are really nice at helping you out