Yes, it is possible to file both civil and criminal cases in respect of defamation simultaneously. The civil case is for claiming damages for defamation. On the other hand, the criminal case (under Section 500 of IPC) is for taking action for the offence of defamation. Both are separate and action can be taken for both at the same time.
Since offence under Section 500 IPC (for defamation) is a non-cognizable (NC) offence, therefore, the police does not have the power to register FIR in such case and it can only register an NC complaint but cannot investigate it, which appears to have been done in your case. You have to approach the Magistrate court with a private complaint if you want to take criminal action in respect of defamation.
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