The problem of legal stay in Georgia if the period of stay in the country established by law has expired (for citizens of CIS countries the period of visa-free stay in the country is 1 year).
If the refugee status is granted, then together with the decision of the Migration Department on granting refugee status you will receive the right to reside in Georgia almost completely similar to a resident permit.
However, while your application and documents are being considered by the authorised bodies of Georgia, you will have a basis for legal stay in the country in the form of a document confirming the fact of application for humanitarian status.