Left by resigning without serving notice period - HR denied that the my data is finalised and absconding letter is sent

I have recently quit my job with an Indian company(X) without serving notice period. I have total experience of 4 years but I worked for this particular organisation just for 3 months. I was working on a project, which essentially needed adequate system, which X failed to provide. The Main Lead raised the question on my performance which made me feel demotivated and upset. Thus one day I just logged in and mailed my resignation to my reporting manager marking HR and left the company.
HR called me later during the day, I explained few things to him and my situation when at work. Now please explain me the consequence on below situation:
I named X into an resume with the prospective employer, will I get chance to change my resume. How does the new employer consider me when they get to know these things or if I am unable to submit the experience or reliving letter. Can I honestly say to new employer, I won't be able to provide all the documents related to X. Is this okay?
Later after a month, I talked to the old HR and accepted that I regret my action. Further made a request to issue the reliving letter. HR denied that the my data is finalised and absconding letter is sent. What options have I got next.
Look forward to your urgent and valuable advice.

Exit was based on emotional reactions.
Such decisions are best avoided.
Now that the act has been done and you compounded it by including X in your resume.Be open and honest to tell your new company that you left after discussion with HR and now they are declining relief papers for failure to serve notice period.
It is better to come out openly and inform your likely employer or in next company do not mention this short employment.
Biggest lesson here is no emotional reactions to quit.
Even if unhappy try and quit on decent footing in future.

Sir, thank you for the quick response.
Will try to convince my prospective employer as you suggested.
But just to add, i tried to work with existing resources and tools which was difficult to complete task assigned. Made request for few tool upgrade, got it rejected upfront. Even after working efficiently, I was blamed again and again in meetings. Thus felt this is an endless project and the management has a very little value to its employees time and suggestions.

Got it. Thanks again.
Surely learnt a lesson being there. Will definitely work on points you mentioned.