Did you try speaking with your new employer? As the situation is broken, you can seek help from a Lawyer. I don't see anything else would help you at this point. Approach a good lawyer and let him speak with your present company. Before all these, please speak with your new employer and explain them the situations and show them the bank statement which proves the salary credit happened regularly and you are a part of that organization, but they are creating complications during your transition.
I have seen employees passing the verification done by Oracle (one of the most stringent employer for employee verification) by supporting documents, while the last employer created issues like your present company.
Send a written resignation(not email) duly signed and dated by registered/speed post(basically to get acknowledgement)
Resign as per your appointment order.
In mean time talk to your prospective employer and enlighten him regarding the relieving problems you are facing.
I am sure the new company will understand and hopefully will not press for relieving letter etc from old company.