Hi,
I also have a 2014, (63 plate), A4 Black Edition, when I first bought it, the tramlining was awful, snatchy, and sometimes scary!
I had Continental Sport Contact 3, changed the fronts for Eagle F1 Asymetric 2, and it's like I have a different car! I'll change the rears soon, but really, change the tyres, Contisport have a reputation for tramlining, would never buy them again, and they just get worse as they wear.