My first thought was not the US States tying them together - but the G theme. So I am thinking a name like Imogen(e) (nickname Immy), Bridget (Birdie), Ingrid (Indy), Evangeline (Evie, Lina), Margaret (Maggie, Meg, Peggy, Daisy), Margeurite (Maggie, Margie, Meg, Daisy, Rita), Magnolia (Maggie, Nola), Marigold (Mari, Maggie, Goldie), or Teagan (Tea, Tiggie or Tilly if an L middle) would be the way to go.
There are G names themselves - Genevieve (Evie), Gemma, Gwendolen/Gwendolyn (Gwen), Grace - but I would use a name with a different starting letter to G and V but with G in it - so all have G’s in middle of name (as with names I’ve suggested above).
Either way, I definitely wouldn’t do a States theme. I don’t think you need to feel committed to that at all. Georgia & Virginia are stunning names in their own rights! Of the two you’re considering, I think Caroline suits the set marginally better but I personally prefer Theodora.