Hi OP - I was in a similar situation a few years ago (job hunting whilst burnt out in an intolerable situation)
@FollowYourOwnNorthStar ’s advice Re self care, nutrition etc helps
Relaxation - eg a bath before bed, when it was really bad I also got a massage every couple of weeks
I was seeing a therapist at the time and that helped, as did a v emotionally supportive partner
I had a lot of quiet evenings/weekends, & baking was another thing that helped me with the stress (bread, pastry etc)
If you were in the private sector, I’d have suggested you sign yourself off with a ‘physical’ illness (D&V or dare I say it Covid) - one which relies on self reported symptoms if you don’t want stress leave on your record
As you are public sector can you not max out the 6 months paid sick leave?
In my case, the toxic situation ended with me being made redundant (I wept with relief!), and I quite literally spent 3 months at home pottering (eg bit of cleaning, cooking, exercising every other day) until I got my energy and mojo back
The job I got after that was much less stressful and this also helped my recovery
Looking back, I wish I’d been more open minded about leaving for a ‘worse’ job as the whole episode cost me several years of my life and I will never get those back