Pay slips, EPF statements, Bank statements, and Form 16 for this Financial Year would all come in handy when the other company is willing is employ you without the relieving letter. Do not assume that these would save you.
But, at the same time, do not rule out that the other company might think that you have left the company without giving any notice or that you are absconding. When background verification does happen, your company might even mention that you are absconding, if a relieving letter is not provided.
Remember, you not getting a relieving letter is your problem and not the other company's. They might tolerate this excuse for a while and later on move on with other potential candidates.
What I would suggest is to reply to your resignation email asking for the release date and cc the HR manager, your department's director, and any possible person who you think can help you out with relieving. If you have made any kind of rapport with any bigshot at the company, reach out to them. They can help you out. Remember to be polite and professional in all your emails.
Leave the company with the relieving letter, FNF statement, and any possible letters which are provided to the other ex-employees.