You didn't provide a link to whatever offer you're referring to, and I can't find that $12 dollar cost anywhere on GoDaddy's site via normal navigation, so there's an obvious speculative element here.
I was able to hunt down this price comparison page via Google, which does list a $12.99/yr offer.
If you scroll past the comparison table, then keep going further past "Those who use" and "About our SSLs", you'll see a very small "Click here for product disclaimers and legal policies." link. And from there:
‡ Annual discounts available on NEW purchases only.
(In case the page I link to isn't the exact one you saw, there's probably a similar link somewhere on your page so look anyway.)
That double dagger symbol refers back to the drop-down where you'd select how many years for which you want to purchase the certificate, and seemingly no other references to the price, if you want to yell at customer service and try to wedge a discount out of them or something.
But short version: No. You're stuck paying full price.
[A bit later]
Interesting. Now that I've been to the offer page, I see this constantly: