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Thank you for writing to Just Answer.
I'm sorry to be the bearer of bad news, but Georgia does not recognize common law marriages anymore, and unfortunately, common law marriage does not apply to lesbian couples. Here is a link to information about common law marriages: http://www.gadivorceonline.com/gapages/Alimony/commonlawmarriage.asp
Is there a neutral third party that could go with you when you go pick up your pets and belongings? It would be good to have a witness, as well as to help keep things calm. If you can't care for your pets right now, contact an animal rescue organization in your area, and ask if they have any "foster parents" that could care for the animals until you get settled. I was able to find a local animal rescue in your area: http://www.adoptapet.com/adoption_rescue/71096.html
It's hard enough breaking up a relationship, but being threatened with the loss of your animals is just cruel. I hope this helps you find a safe home for them until you can take them with you.