2.If this does not work and you were paid by cheque, then obtain record from bank and state facts that you not have other documents like relieving letter/ offer letter/appointment letter but have this proof of payment
Best of luck
2. Look into your personal mail box, if you did not copy your official mails at all. You would definitely find something.
3. Form 16, 26AS statement, PF passbook, Bank Statement could be another resource for you.
I do not think its not possible to have access to a single documents to prove that you worked in such an organisation. In the interim you can ask for a conditional offer letter that within a certain period you would provide a concrete information to your new employer, failing which they reserve the right to terminate you.
I am sure if you have an edge over your competitors and the new employer liked you, they should agree to your request.