Bumping an old thread in shock to see that this has been a problem since 2013 and nothing has been done to fix it.
Picked up the game recently after not trying any of the new generation NFS. Had the same audio problem after installing patch it was so annoying that I literally could not continue playing the game after 15 minutes. Simple HDMI cable to a Sony TV. I can confirm that if you switch to stereo instead of Dolby 5.1 it fixes the problem but this is a huge inconvenience as this changes the sound to everything on your Xbox 360 system.
Man I am very disappointed in EA for simply not taking care of this problem after so long. Definitely not going to go buy any of the other Need For Speed I missed out on after this experience. Really shows how much they care for their games, I'm sure its a simple audio glitch that they could have fixed.