We got a 'no more tears' 'won't sting your eyes' pretty bottle of shampoo and pointed out all the features. Dd still doesn't love it, but at least there is no more screaming. Maybe role playing and washing a doll s hair. might help.
Colored water in the bath, water balloons with coloured water inside are good distractions- pop the balloon and the coloured water comes out.
I wonder if she has bigger issues with sensory processing going on that is making this really hard for her.
My son responds the same way to baths and hairwashing and he has sensory processing issues along with his autism. For him we were advised NOT to get him to put his head back- do not force him into a lying down position and stop trying so hard to coax him into putting his head back while washing. It is a very disorienting position and if the issue is struggling to understand your body's position in space then it will just raise anxiety and make hairwashing even harder.
Things that work for us are giving him a stack of warning that we're planning on washing hair. We use a visual calendar and we have a picture for hairwashing that we'll put up and point it out for the week.
We usually do that bath and wash hair earlier in the day while he's still fairly fresh. We bring a large mirror into the bathroom and make sure he can see everything that's going on while it's happening. We go super slowly and calmly and have both parents there so one can be holding his hand and helping remind him he's okay and to relax and to watch the mirror. We involve him as many choices as possible so he chooses the size cup to pour water with and whether he wants it poured on slow or fast etc. We keep a towel right there and dry his face instantly after every cupful. Just whatever we can to take the stress away and put him more in control. It's made a big difference over time in that he smiles and feels proud about getting through hairwashing, rather than screaming and panicking now.
As for bathing- we don't bath him everyday and I think a washcloth is perfectly fine for a few days. We only bath him about twice a week, though we bath his brother every day so we always offer the option and try to make it interesting. Often putting his favourite toys into the water and making a game with them that he can't resist will entice him in.
Sounds like you have a fair way to go in regards to teaching her to get her hair wet. Does she have fun at the pool? Swim classes are good for teaching them gently to get their head wet.
BTW, I don't think you or she is disgusting at all. In fact, the hair wash screaming is not a small part of why our family is shampoo free. Mine don't mind getting their hair wet with water though, so they get a rinse in the shower or bath most nights. I ran out of shampoo around last October and it has been fine (except one unfortunate incident of lip gloss though the toddlers hair at christmas time but gran had some shampoo with her).
I don't think not washing hair is gross, I only wash my kids hair if they go swimming to get the chlorine out. But the cradle cap does sound like it needs to be dealt with, and she needs a daily bath to wash everywhere else. What I would do is make the bath as fun as possible for a while until she gets used to it. At least then she's getting washed and you can then try deal with the hair.
You could try:
Food colouring in the water.
Cut out craft foam shapes she can stick on the wall and make things (I have done blue water with white foam 'icebergs' and sea animals, or pink water and cut out flower shapes to decorate the wall, etc).
Bath paint, crayons.
Even use washable paint yourself and paint the bath and make a little world with figurines to play with and no water and let her play in it during the day and get used to the bath.
Glow sticks in the bath.
etc.
ETA- I just googled and there is a whole website called 'bathactivitiesforkids' with tons of ideas.
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