You should have receipt of your resignation with you with date ofits receipt clearly visible on it.
You may send your resignation to your appointing authority or Managing Director, and express your wilingness to to coperate with the company for smooth handover of the charge (specify a date within maximum of 1 month)and to pay the notice pay for the balance period. The good office of MD shall be able to provide the relief whcih your line manager is not willing to due to their limitations.
The new policy which is 5 days old is announced to you after your resignation as per terms of employment in appointment letter is received by your company.
Usually the circulars/HR communications are supplied in big companies by emails/attatchments thru Dept Head and then line management on Pan india basis. You should keep copy of circulars safely to establish the practice.
You need the persuasive skills to explain the situation and convince the team at new company .
The team at new company is also aware of the trends in the industry, and problems of attrition/vavancies etc and can help you.
They can employ you on the basis of copy of the resignation,proof of its despatch,receipt/acknowledgment and may ask you to sign some documents/declaration.
You may also persuade and convince old company that beased on the exisiting terms of employment in your appointment letter you have finalised your new venture and you are willing to coperate with the company to handover the charge smoothly upto maximum of one month( specify the date)and by paying the notice pay for the balance period. You can provide the references for some good candidiates, to fill the vacancy with your exit.You should submit all company property under receipt.
Your line Management are leaning towards the 5 days old policy, since they have to or they shall face the compliance issues and If they have not supplied the policy in time they are at fault.
However you need them therefore mainatin good rapport.
With your efforts and facts you should be able to manage the situation.
You can consult your local lawyer with all the documents .