We haven't had the bath thing here, but I have a nephew who totally freaked out (and I mean FREAKED lol) when he was 3-8 years old if any food that was remotely non-white in color appeared anywhere near his plate, or if any of his foods touched on the plate or if.... oh my.... anyone dared put bread on his plate with the crust still attached. The first time I ever met him, he was two and he had a thing about car seats - he didn't like the way the arms of the seat hugged his body. We had to ride together in the car for 20 minutes and he screamed non-stop the entire time... which was apparently typical for him.. in a car seat. He was also not fond of sleeves, and refused to wear coats in mid-winter. Some of it may have been sensory, or maybe it was just quirkiness (I don't know his IQ but I suspect it's sky high based on what he's accomplished in school and based on conversations with him). It wasn't anxiety and it wasn't something that was just the typical type of reactions a parent could help the child work through.
So I can't help with the bath situation, but I can let you know - my nephew grew up to be just absolutely the coolest young adult ever! And he's so mellow now too... and he even eats foods of different colors, wears coats in winter, and doesn't scream in the car. He outgrew most of the quirky stuff at around 9 years old, and fwiw my sensory dd also outgrew a lot of sensory-related quirkiness at around 8 years old.
polarbear
