Not serving notice period

My name is Raj Sharma and I joined one organisation on December 2012 and left them on June 2017 stating them few health issues. Also, they had very less business and I was working on same platform where my novice teammates were working. I had job insecurity. In the meantime, I got new opportunity with an immediate effect and I joined them. Now I went back to my old employer for documentation I.e Releiving Letter and Recommendation Letter. I also agreed to pay them my one month basic salary in lieu of notice period. But HR is saying that they will write in my experience Letter about not serving the notice period and due to some health issues I resigned. Honestly, that letter will not be in any use if they write so. They are harassing me and now saying that tell your new employer to send us an email about your joining with them and we will give them your back check and later we will give you documentation. The contract with my old employer states that it's between them and me. Either the party has to serve the notice period of 30 days or one month salary in lieu. I also don't want to reveal my new employer to them as it is my personal interest. Kindly suggest me what should I do to get my clean paper work.