I don't think you can expect a few weeks to make a difference. I'm not saying it will not help but just that you cannot expect and plan for an improvement.

I think you should make your job change decision based on which job will have the least stress on your life. Taking some time off to get settled and prepare for your new job would be worth something. I don't know how FMLA will make any difference. It is primarily used to take time off without a risk of losing your job. If you have any paid personal or sick time available, you should consider using it up if you can before you leave the current job.