Genderless Pronouns?

I am currently writing a book in which my characters constantly switch bodies, using both male and female genders. It would be inaccurate to refer to a character as only “he” or “she” since each the idea is that a character can inhabit both a male and female body. Should I set a single gender throughout the book for each character regardless, use new awkward genderless pronouns, or alternate between pronouns based on the body that the character inhabits at that moment? Open to other suggestions.