I got a job for which I applied before joining present job

If the establishment that you want to join has requirement of NOC then apply for it under proper acknowedgment.

Explain the facts and reasons in writing under proper acknowedgment to both current and future employer..

At the max, if there is shortage of time.. write amn application of NOC stating date of original application to the said job to your current employer and get acknowledgment of recieving of such letter from your present company.

If you get NOC within time all well and good .. other wise while verification of certificates you can show that you are working as JE from certain date based on offer letter, joining report, ID proof and copy of the recieving of your application for NOC. This shall suffice.

Anyways, you first asess whether al this is necessary i.e. showing that you are working currently as JE would give you edge for selection in State Govt post or not and take decisoin accordingly.

Yes you may write under proper acknowledgment.