Okay guys! I fixed it on my PS4 last night! I don't know if this will work for the Xbox but it worked for me.
Turns out the game was not fully downloaded even though my playstation 4 said it was fully downloaded, installed and started the game.
When I would go to the home screen I had a little "83 percent installed" circle or something like that in the bottom right of the screen, despite the download screen on my ps4's home screen telling me it was fully installed.
To fix this, I turned off the internet connection to my PS4 and went back to the downloads section and noticed there was a little segment at the end of the Need For Speed Payback download that wasn't filled in even though it said 100% installed. Then I turned the internet back on and went back to this section and suddenly it started downloading more of the game. Took almost 23 minutes to finish the download, then I went back into the game and started the Revenge mission again. At first everything was all low resolution the suddenly the announcer came on the radio and all the textures loaded in to 1080p. Haven't had a single issue or low texture problem since.
I hope this helps!
Let me know if you guys need more details on how I did this if that doesn't make sense.
