Can I deduct TN state and local sales tax for the 7 months I’ve lived here and still get my GA state income tax refund for the 5 months I lived in GA?

Yes, if you are due a refund from GA, you will get that refund. 

I read your other question and I think what the program is saying is this:

If you claim the GA taxes paid on your Federal return this year, NEXT year you will need to claim the GA refund as income, BUT if you claim only your TN state tax on your Federal return, you won't have to claim the GA refund as income next year.