If your HR is behaving in a non-professional manner, try to speak to your manager and unit head before speaking to HR. Also communicate the issue with your new employer as well.
In case HR is pressurizing you to wait some more days, get the new relieving date in written (mail or paper) and otherwise, get the attention of the CEO
12th March 2019 From India, Bangalore
I agree with Consultme.
Your notice period counts from the date of acceptance of resignation. If your HR has not accepted your resignation you can not claim that they are not relieving you. Of course no company has right to hold an employee from opting the employer of his choice except under legal , fair and just agreement if any.
Before accepting the condition of new employer you should have cleared with present company and thereafter only agree on date of joining in new company.
Anyways now all you can do is request and pursue with existing company.
Wish you good luck !!
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12th March 2019 From India, Mumbai