More work is never the answer to burnout. If you are burnt out, it won't get better. Even if something happens to change the current situation, it wouldn't feel as good when you are burnt out.

I was in a very similar position during 2022. It's a perfect burnout recipe, chronic stressors everywhere and no power to change them. I could feel it building, but was still able to work and the personal stakes were high, so I wanted to make it to the end (which would have been in early 2023). I didn't. I had a break-down in fall, I couldn't do anything anymore (no focus, no energy, really bad memory, lots of mistakes at work). I took sick leave and have been in recovery since, and still don't feel really alright. I have met people who went further than me and have done themselves really bad, and even physical damage doing so - psyche and body feel different, but they are parts of the same organism, and to overtax one is to overtax the other.

In no way was it worth it. I would have chosen differently if I had known what the later stages of burnout felt like.

I really mean it - if it's in any way feasible and you've exhausted other pathways to change inside your role, quit before your body does it for you.