As a hiring manager, I can often tell when someone is burned out the way you describe - and it makes me question a lot of things:
- Why did the candidate stay in a role they so clearly hated?
- Will they still be burned out if they come work for me? (The answer, by the way, is almost always yes because burnout takes a lot longer to recover from than most people allow themselves between jobs.)
- Since they're likely to start working for me while burned out, how much productivity can I truly expect from them?
- How long will they manage to work for me before the leftover burnout gets too unbearable, and they quit? (Probably not very long, based on experience.)
If you're only staying in this role because you're afraid of potentially awkward questions during future interviews, then you have two options:
Stay but prioritize your well-being
This option is difficult. Speaking from my own experience, you're unlikely to recover from burnout as long as you remain in the situation that burned you, and highly likely to get more burned out. If you choose this, @A.S.'s answer has some good suggestions for things you can do to make it fractionally less miserable.
I would also recommend using every minute of PTO you have. From your description, it doesn't sound like you have meaningful work to do - but even if you did, your health is more important. So take off as much time as you can for the remainder of your stay at this company. This will at least moderate the effects of the burnout by removing you from the situation as much as possible - and it's not like you have a reason to save that PTO anyway.
Another possibility would be to investigate whether your company offers leaves of absence; or if there's no official policy, whether your manager might be willing to arrange one. It might feel a little awkward going on leave for a couple of months, then quitting for a new job shortly after your return, but again: you need to do what's best for your own mental health as long as you plan to stay at this company. You might be burning a bridge, but given how you describe this place, that's not a big loss.
Finally, keep in mind that getting a new job usually takes time, even in a candidate's market. Assume it'll be at least two to three months between when you start looking, and your first day at your new job, counting both the job search itself plus as much downtime as you can afford in between jobs. So if you do intend to stay at this company for one year, start looking for your next job at your ten-month mark.
Leave now and get yourself into a better place ASAP
Having a single short stint on your resume isn't going to hurt you in the long run, unless it's your first job. All the issues you list in your question are extremely valid reasons to quit a job regardless of how long you stayed there, and unless you have a pattern of job-hopping every six months, most hiring managers won't be concerned by a six-month stay at such a miserable place.
If you can afford it, give yourself at least two or three months off before diving back into the job market. Again, hiring managers can often spot burnout in candidates, so you want to make sure you're rested and recovered before diving back in.