Date of joining and offer letter - offer was shared without the joining date

Hi I have got an offer from a company,

My current company has a notice period of 3 months and my 3 months are about to get completed soon. I have recieved an offer letter from the company through reference and hence the interview process was smooth. Since the new employer knew about my notice period he still asked me to negotiate the 3 months to 60 days. The offer was shared without the joining date and just the salary breakups and positions. Now I am chasing the HR again and again for the revised offer letter with doj. Everytime I call he/she says, it would be shared shortly. I hardly have 15 days left and I still dont have a revised offer letter.

I feel like as if I am begging for offer letter. I had offers but since the reference is of the closest and the position is good I dont want to miss this chance.

Please help me understand how can I handle the situation.

Dear Colleague,
You may write a mail to the HR. Kindly top up the offer letter mail and write a mention that you will join on such and such date in a polite manner. Keep copy for your reference. This is what best you can do when there is no response from other side. join on the date committed in your e-mail.
God Bless!!!