Offer Letter/Releving Letter

How important it is for any company to provide a new joinee with an Offer Letter.
Is there any kind of act regarding the same.
The company where I am working does not provide any proof as to mention the joining date or the designation or any such things.The pay slip is the only thing that can be shown.
Which is not accepted in most of the corporates.
Also the company has strict rules as to provide one month notice in order to receive releving letter.No releving letter will be provided in case of short notice.
There is lot of manipulation that is done here in respect of salary and all.
I got a new job in a better company than I am currently working.But I am falling short of one month notice., hence will not be getting releving letter which i need to show in my new company.No amount of requset seem to work here.
Is there any thing that I can do ?

do a thing you have been provided a outlook from your company rt?
mention that my name is so and so i am working with you from date .. month ... year (mention date ) along with your designation and say would like to resign from the post due to my personal reasons. and forward to your hr and put a cc to your persona mail account
later take a print of it itself will be a proof for your joining date and resignation letter