Regarding not getting NOC from employer

Until unless there is a bond submitted by the employee at the time of joining or there is a contract that he cannot leave before so many years or if he leaves then he will have to deposit some amount before leaving the present job, etc, the organisation or the department cannot refuse for giving NOC to the concerned employee.
Now coming to your case it is clear that there is no such bond or contract signed by you at the time of joining. So it is clear that your employer should provide you NOC.
In such a situation what is to be done is that you please send one advance application form duly filled along with all the necessary documents to the authority where the new position is advertised. You have to mention on the top of the application form that it is an advanced copy and through proper channel copy will be reaching them when your department forwards it or gives you a NOC.
Your new organisation will call you for the entrance exam or interview whatever it is and will keep your selection under withheld category until you produce a NOC. You can take leave for urgent personal work and attend the interview or exam whatever and whenever it is taken.
By the time if your organisation doesn't forward your through proper channel application and doesn't give you a NOC then you appeal to your higher authorities for justice and clearly mention that you have already been selected in the new place and it is a question of your career and justice should be made to you by releasing the NOC or forwarding the through proper channel application. if you approach the higher authorities, they will definitely help you in the matter.
Please remember that if your department is not giving you NOC just like that based on some insignificant matter then law will be in your side so you should not be afraid of anything and go ahead with your career making.

Is any enquiry going againts the sleeping case against you or they have just given a warning letter only? In any case, they should not stop giving you the NOC. If any contemplated departmental Inquiry is there, sometimes the management may withhold issuing NOC. If any Vigilance Investigation is in progress also there may be some delay in issuing the NOC. If no enquiry is going on or not contemplated, the management should not say no to issuing NOC If necessary they can say subject to the position at the time of resignation. They can forward your application.
If they say no to that it is against the rules under article 14 of the constitution. You can proceed legally. But this is the last resort. You send an advance copy of your application directly to the new post mentioning that you have submitted your application to your department for forwarding the same with their remarks. Then you will get some time and you try to convince your higher-ups and try to get the NOC and see that they will forward your remarks. That will solve your problems. You can try to use any influential connections to get the work done by your bosses. Something may materialise before you get the selection. If nothing is happening and your new employer is insisting on NOC to issue an appointment letter. you have to take the help of a lawyer and file a case in court. Generally, if there are no concrete charges against you, no court will give a verdict against you.

You have mentioned that you were issued a warning letter for your indulgence in sleeping though for a short time in your shift. In this case, management is always at liberty to take action against the errant employees. It is definitely not a major offence and even no explanation was asked from your side.
You have to take a little pain to assess the real developments against you regarding the disciplinary proceedings. If it has not taken so far, take your immediate boss into your confidence convincing him that on that night, you did not have enough rest in the day time and as such there was the spell of short sleep. Your boss will judge you in terms of your performance and your general behaviour. It could be possible that you might be exonerated from such a malicious charge and if that happens, you will be obliged with the NOC. You should be rather be tactful so that situation turns out favourable in your case.
You cannot apply afresh being a fresh candidate for the post since suppression of facts leads to violations of service rules attracting disciplinary actions.