You are correct, it is my fault that I did not provide the proper notice period while getting the new offer. But, I am already willing to serve 1 month of the notice period and buyout the rest. The company has not agreed to it.
As for my options, I too would prefer an amicable solution where all 3 parties come to the same conclusion. But I am trying to weigh my other options as well (legal or otherwise) in case my current company remains adamant about not honoring my request. Thanks for your tip!