Job Ethics Question | Switching jobs & Leave

I have been offered a new position at the collegiate level coaching and teaching a few grad level classes. I currently coach my school's HS team and teach middle school so it is a great opportunity and move up. As a teacher, we are allotted 10 days per year for sick and personal use. I had accumulated about 50 days in my years here as it leads to early retirement if you save a bunch.

The collegiate position, however, is not in the same retirement so those days do not count for anything. I spoke to HR last week and they said the policy for my scenario is "use them or lose them." I asked if I could be paid out or come to any workable solution, but they said no. So, I have been liberal in my use of sick days since and used twice in the past week. My principal and now HR are hounding me on maintaining attendance. Legally, they have no leg to stand on, but I was wondering what others thought? Teachers have told me to ignore them because they cannot do anything, but I was wondering if I should take into account anything else? But basically, it is hard to motivate myself to work for free.

For eleven years, I had come to work sick, missed personal opportunities for trips with friends and family, etc. and now being hounded to not use them feels pretty shitty to me. However, in the same breath, they say they are very sad to see me go.