Initial adhoc appointment and 2nd govt job.

You can resign to the present job and get the resignation letter approved and get the relieving letter from the present employer and join the new job. For joining the new job no NOC is required as you were not in the job when you have applied for this job. But in this case, they may not consider your experience for counting your service because both are government jobs and there is NOC.
If you have informed the present employer at the time of joining that you have applied for another governemnt job. If so you can ask them to give a NOC certificate so that your expereince will also be counted and continuation in-service conditions will also apply.

As you mentioned that at the time of application and test for the job, you were unemployed and only during the interview time, you were employed. It is known fact that everyone would like to start working as soon as possible and everyone will understand the scenario.

As you are keen to take up the new job(government) , you will have to put forward an official resignation letter in order to get relieving letter from the organisation and if you have not completed 3 months which is the probation period, you can get the release very easily but if you have completed the probation period, you may require to complete the formality of working for 1 or 2 months as per the requisite of the company or organisation.

It is always better to complete the official procedure in the required format of the company. It will help you get your required documentation and other relieving procedure with ease. Kindly follow the company procedure for quick and easy clearance from your employer at the earliest.

As the joining in the first job happened after applying for the second job you are not responsible for the NOC right now but you have to resign from the first job in order to join at the current one. You have to find out the terms and conditions of the first job and see whether you have time to give them notice for a period of one month for resigning. Generally one month period is there for notice period in such cases. If yes, you can do that but if time is not available then you will have to resign with immediate effect for which you might have to deposit one month salary along with your resignation letter. If you do that then you will get a relieving order and that can be very well shown in the current place. As you have not applied through proper channel, they might not ask you for a NOC. They may take you as a fresh person without any earlier job. This may create some problem in your provident fund etc so better disclose this whether it is asked or not.

In such cases honesty is the best policy and you will not have any inconvenience or hassle at a later stage.

As you have stated that you did not have any any job initially and only during the process of interview of the first job, you got an offer of the second one. With your submission, you joined the 5 months ago and under that case, you will have to ensure that whether you are under the probation period or not. In case, you are under probation and any bond is attached has been furnished for the non - fulfilment of discharging for the said period, you would have to submit the bond money coupled with any money for being defaulter in rendering your service in the present organisation and then only your resignation would be accepted. You will be provided with a relieving letter so that you can join the second job smoothly.
However, if you have completed the bond period and have discharged your duty for the period as stated in the joining letter, you can submit your resignation and upon approval of the same, you can join the second job.
In your case, NOC is not applicable since you did not render the first job of sufficient length.