Why can’t you do both? You can line up a new job and take a break in between. I regularly take a month or more between jobs. Most employers are fine waiting a bit longer for top talent.
When I went through something similar it would have taken way longer than a month to recover. I didn't know how long when I took the break. Also I was too burnt out to interview well.
The stress from not having a job lined up would have eaten me alive. I wouldn’t recommend that for anyone. I’m glad your case panned out but I would tell OP to line up a new job if they can.