OK a little complex. English is a beautiful and complex language, and there is a lot of flexibility in it.
"a woman's beautiful eyes' emphasises her eyes.
" A beautiful woman's eyes" emphasises the woman, the person who has the eyes.
"Trump's 2 top aids" emphasises the best 2 aids he has. "2 top Trump aids" could be any 2 people, but "T's 2 top aids" means the very best 2 he has. It is specific.
"I just want to my Genuine Georgia's Best Peanut Brittle". should be "I just want my ...." i.e. I just want to eat this food.
You only have an apostrophe after Georgia, the person. The word, Genuine, is like a description, an adjective describing Georgia or the food. You only have one apostrophe, so it is after the person's name.
I don't know the food. I would say that Genuine refers to either the food (i.e. it's the genuine food of Georgia) or the person (i.e. the genuine Georgia, this Georgia and not another person called Georgia). It also seems that the whole name is a brand name, a trade name. So Genuine Georgia is the company, and so has an apostrophe at the end of the (entire) name.