My ITIN W-7 request got rejected. I have reapplied (with notarized document this time) and am hoping that I will get my itin number and the request will not be rejected again. But I may not be in time to file my Georgia state return. I owe taxes on my Georgia state return and am getting a refund on my federal return. Will I need to file an extension request for my state and pay estimated tax OR can I file my Georgia federal return indicating "ITIN applied" OR is there any other option available?

Make sure to pay Georgia taxes by the due date even if you don't file the return.
You can file the return only after you get ITIN. You should apply for extension.