Sorry, I should have made clear that the broken codes were games purchased after the site update. The ones purchased before worked fine.
samlii: The full code is also actually in the html source as data-giftcode on the item-row div for the gift code. If someone has time they code make a user script to correct this.
This is the solution. Well, I'm sure the codes in the emails work too, but I tend to delete those pretty quickly. I was able to extract the code from the page and test it, and it does show the game available for redemption.
Thank you samlii, and everyone else who posted solutions. I can use these workarounds to get at my codes until GOG fixes it proper.