Advice needed regarding moving jobs between local authorities!

Hi all! I was just wondering if anybody has had experience with a very specific situation. I'm changing jobs and will be moving from one local authority to another. I think I've read somewhere that your annual leave allowance etc follows you between roles. Does sickness also follow you?

I'm currently experiencing severe burnout (I had 2 weeks off a few months ago because of it) and had 1 day off at the start of the year after catching the flu that had been going round the office. I'm considering putting in a sick note for some of my notice to give my body and mind chance to recover and get both my personal and professional life back on track, however I don't want to start my new role with 3 instances of sickness on record (my sickness record is usually squeaky clean for years). Does anybody have any experience around this?

TIA!