Each of my party members is carrying a blanket and yet I catch illnesses (mainly the one where it says their head feels stuffed with cotton) no matter where I travel to. I'm just trying to follow a guide but I end up getting my whole party sick long before I make it to the first major location. I feel like I am missing something basic but I don't know what it is.

One time I was lucky enough to make it to the Dwarven Pit and the healer wanted an exorbitant amount of money to cure the disease! What am I supposed to do?

Your characters have a talent called "treat disease", use it. The diseased has only one attempt to get healed by another party member, so save first (it takes 24 hours until you can retry). Put the herbs you have into the healer's inventory, different diseases require different herbs.

Praying for a miracle at a temple of Peraine or buying a healing elixir from a herb merchant will also work.