So I'm debating whether or not to go back to the store and see if I can get the game code, or ask AMD here if I could get one. Any other thoughts what I should do?
I you really are that adamant about receiving Civilization VI for free via the promotion, then you're most definitely better off returning the GPU for a full refund and then purchasing another one. Otherwise, even if you exchanged it for another RX 480 at the same location, the purchase date would still remain the original purchase date, not the date of the exchange, thus you wouldn't qualify.
As for you receiving one from an AMD employee here in the forums, sorry to say, that simply wouldn't be possible. Either way, good luck with your return and repurchase if you decide to go that route.
Back to the store I guess, what are T & C's of promo ?.
Unfortunately we don't have spare game codes, so you might be best served by returning the product and purchasing it again.