If not, then start searching and leave when you have found something you'd be happy with. Having been in a burn out situation my self, I'd give my 2 cents and recommend maybe downgrading in responsibility to give yourself some time to heal (some people heal through doing a lot of very fun but hard work, so do what is right for you).
Staying on for too long, especially if it has impacts on your health is the wrong option if others exist.
If anyone asks in later interviews why you did not stick with it, simply explain that you did try to change the culture but it simply would not stick. Explain that you are a professional, and would like to work with other professionals of the same caliber.