The issues: bishul/ heating, decree against bathing in hot water and possibly cold water, sechita/ squeezing liquid out of a solid (even hair)
The solution: If one is extremely uncomfortable one may take a cold shower on Shabbos. One should put shampoo on one’s hair before wetting their hair and then rinse it off without touching one’s hair. Liquid soap may be used.
On Yom Tov, some do not allow taking a hot shower but others allow.