he wanted me to . . . buy out that notice period.
I'm not sure what this might mean.
I have not signed on any contract/offer letter with them them but they are saying company has mentioned it in offer letter.
Who are "they"/"them"? I ask because, in the first sentence of your post, you referred to your "old employer" as "he." To whom does "company" refer? When you say, "company has mentioned it," to what does "it" refer?
you signature does not matter since if you dont decline or raise concern this means you are accepting it.
Just wanted to understand - what my rights are as an employee here to get my final paycheck and relieving.
I have no idea what "relieving" means in this sentence. As far as your final paycheck, California Labor Code Section 202(a) provides as follows: "If an employee not having a written contract for a definite period quits his or her employment, his or her wages shall become due and payable not later than 72 hours thereafter, unless the employee has given 72 hours previous notice of his or her intention to quit, in which case the employee is entitled to his or her wages at the time of quitting." In other words, because you gave more than 72 hours of your intent to quit, you're entitled to your final paycheck at the time you finish your last day of work.