At its root, I don’t think this question is really about the logistics of taking a sabbatical.
Something I didn’t fully recognize when I was burnt out: burnout is a personal emotional issue. At that time I (mistakenly) considered emotions a bug and not a feature.
So whatever direction you decide to go on the sabbatical, I highly recommend endeavoring to discover some emotional support and growth.
Online talk therapy has helped me greatly, but mostly the book I wish I had back then is “The Happiness Trap: How to Stop Struggling and Start Living”
https://www.amazon.com/Happiness-Trap-Struggling-Start-Livin...