Can I take my Georgia state exam if I don't have a sponsoring broker?

Yes, you only need a sponsoring broker to apply for an active license.
You can apply for your real estate license at the PSI testing center, right after you pass the State exam.
If you do not have a sponsoring broker at that time, you can apply for an inactive license.
You also have up to one year after passing the State exam to apply for your license.