If you are making large scale changes to your underlying infrastructure code then The Mikado Method might be useful. Methods and Tools did a review on it.
There is a book on it Mikado Method for Legacy Code by Ola Ellnestam & Daniel Brolund
Another book Working Effectively with Legacy Code by Michael Feathers is also a useful resource.
I've seen this happen with design so many times. It's a structural problem with how people and teams are organized. Now, I feel like I should say that cross-functional teams are not required to be agile. Scrum does require them, but I don't see that you are specifically using Scrum. That said, the structure of "Design team creates some design and the ...