I have had applied for my spouse's ITIN with my federal taxes about 4 weeks back.
I have not received my spouse's ITIN yet.
Do I file an extension for the GA state tax? Or do I file GA state tax stating "ITIN applied for" just below the SSN place holder in the form?
I am on H1B and filing jointly.

If you have a refund from GA, no extension is required; it's too late anyway.

Just wait for the ITIN, then submit the GA return when you get it.