Leaving job after probation

Dear Sir, I joined this school on 15th May 2023 as a teacher for senior classes. Within these 15 days I sought leaves 3 times for gynae problem which was turned down. I am 47 yrs old. I had to push myself to school although I was not in a position to move. I was forced to come to school. On Friday another school finalised the offer for Vice Principal with a better package for which I had said yes. When I informed the previous school that i would not continue for health issues the lady director wants me to put up a formal resignation with all the medical reports attached. My appointment letter mentions my salary only. No Joining letter was given. No service rules were signed. No agreement was signed. No leave rules mentioned except for an official email which was given to all the new joinees. Nothing mentioned about the notice period in the appointment letter. It was 15 days probation was verbally told. My question: Should I put up my resignation? If, can I resign on any other grounds? (I don't want to mention my health grounds.) Is it a rule to attach all the medical certificate/reports? I have got these 15 days salary also. I have to join this new school from monday onwards. Specimen books were given to me without any signature. Can they put me in trouble if I don't tender a resignation? Please help. I m very worried. Regards RS