Good afternoon, Rabbi,
I am an Ashkenazi Jew living in a tropical rainforest. Sometimes on Shabbat it is extremely hot and humid (35-40 C) to the point that I cannot concentrate on Torah learning or feel unfomfortable, interfering with my oneg shabbat.
We Ashkenazim have a minhag not to bathe on Shabbat, but might there be room for leniency here? I have no hot water heater (no need for it) and can easily air dry without using a towel.
Being that the water is not being heated up on shabbos, then shabbos would have the same halachos as taking a cold shower on yom tov. Although it is the minhag not to, but if you are very uncomfortable, as on a very hot day, it is permitted. You should however be careful not to squeeze any hairs. You can use a towel to dry the parts of your body that don’t have accumulations of hair, i.e. stomach, back, hands and feet, but you have to be careful with using a towel for your head. You may place the towel lightly on your head without squeezing it against your head.
Regarding using soap, a solid bar may not be used, and regarding using liquid soap, it depends on how think it is, and it is controversial how thin it has to be.
Igros Moshe O:CH 4-74 Rechitza-4, ibid 75, Minchas Yitzchok 6-32(5), Shmiras Shabbos Khilchoso 14-11, ibid ftnt. 66, Chut Shani 2 pg. 62, The Shabbos Home pg. 495, O:CH 326- 10, M:B 326-30.