I've explored my old save files and narrowed this down a bit. Act 1 spoilers below:
The bug is not present immediately after defeating Alexander -- my main character's wearing the Corrupt Tyrant's Helm in my file after the fight, and if I remove it the warm effect goes away normally. I'm assuming I boarded the Lady Vengeance while wearing the corrupt tyrant's helm, but I can't remember for sure.
My next save after that is on the Lady Vengeance near Reaper's Coast (this is after solving the mystery on the ship, the big ship battle, and the ghosty realm stuff.) In that file, my MC isn't wearing the corrupt tyrant's helm, but equipping it/removing it leaves her with the permanent warm status effect. So right now my best guess is that somewhere on the ship something happened to my MC that makes her receive the warm effect permanently.
