I am a student attending a GA college. My permanent residence is in another state. I earned less than 5,000 in GA, but it was about 39% of my total income. I am claiming a credit on the other state's tax form for taxes paid to GA, but it tells me I have to attach a GA tax return. I don't think I am required to file the GA tax return, but if I do I will owe them money... what should I do?

You have to file a GA tax return to document the taxes you owe GA on that $5K of GA-sourced income. If you owe GA more money because not enough was withheld by your employer, that just means that you owe your home state less (after the credit).