My job is going terribly, but should I quit?

I’ve been having a terrible experience at my new job; it’s been almost 5 months, but feels like it’s been 2 years. My client raises his voice at his staff, has unclear and changing expectations, requires so much work that I always work late nights and through almost every weekend, and will complain to my boss about small things. My boss has been on my side but I worry about how long that will last given the frequency of this guy's complaints. Keep in mind, my 6-month review is still a month away! The conversation about how best to handle this client, responding to his complaints, etc., has been a conversation I’ve been having with my boss for more than 4 months now. While things have been on the upswing recently (as they have been before), overall I have no reason to think things will change, and I really have no interest in rescuing this situation. My confidence is shot and I’ve pretty much stopped taking care of myself—I used to love eating well and exercising, but those habits have evaporated. I have no appetite, I can barely get out of bed in the morning, and I’m almost nauseous when I open my email. There’s an FGB article about 18 (or however many) signs your job is taking over your identity…all but 1 of them were spot-on for me. A few weeks ago, I told my boss I don’t think this client and I are a good fit and that I’ve considered applying elsewhere (spoiler alert: I’ve already done that), but don’t want to leave the institution, as I do love its mission. He said that unfortunately there’s no other spot on his team and to give it another few months with this client. Well, on Friday I got a job offer to work for an agency. It would be an 11.5% pay increase, although they seem disorganized, they’re a bit pushy about when they “need” me to start, and there’s no guarantee it’ll actually be less work, and the work is a bit boring. But it would be getting away from this client. I do wonder, “will accepting this job offer get me closer to my goals?” but —and here’s the frightening thing—I don’t even know what my goals are anymore, or if I even want goals. I'm burnt out and at this point the possibility of making a good paycheck for boring work while having time for a life again is pretty appealing. I also don’t want to burn bridges with my boss, who offered me the job because my former boss (my mentor) gave him good feedback about me when I was in my old department. And I don’t want to burn bridges with her either. Has anyone encountered this sort of scenario before? How did you handle it?