I recently read the BuronOut wikipedia page https://en.wikipedia.org/wiki/Occupational_burnout because I was having some thoughts about it myself. Burnout is a real problem in the software industry. With the exception of 2 months hiking leave (about five years ago), I never stopped working since after University (actually also during). I found that multiple short intensive vacations (long weekends 4-5 days), in times of stress, eases the tension a lot. It also depends how much support do you get at work (from you peers/manager, can you openly talk about it)? And how much support do you get at home from your immediate family.
A few things to remember - once you got a new job it is ok to negotiate the start date. Buy yourself enough time to recover, knowing that you are going back to work, but not now. Depending on your negotiation skills, if you can't get the salary you wanted, try to trade it with vacation days (so you can take more long weekends). Some burnouts could take a few months to recover from. Take your time.... Looking back at your career 10 years from now, these few months are very short, and who knows - you might get more offers during that time... going to meetups, messaging friends for better job offers, asking them about the stress levels in the place they work.