360 logs out when trying to play disc

Very sorry to hear it. However, it sounds as though you self-repaired your console by replacing the disc drive yourself.

I'm afraid no one here will be able to help you at all now as self repair is against the ToU of the Live service and discussion of self repair is forbidden on the forums.

Not even phone support will help.

Your only option now is to purchase a new console.

Hey there cmcardinal24, are you able to reconnect to Xbox Live after getting disconnected?

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I will recommend clearing your system cache, then deleting and redownloading your Xbox profile.

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Here's the links on how to do both:

http://support.xbox.com/en-US/xbox-360/settings-and-initial-setup/clear-system-cache

http://support.xbox.com/en-US/xbox-360/settings-and-initial-setup/profile-move-delete

I've noticed that it is just immediately after playing an Xbox 360 disc. Once I put in an Xbox One disc it sounds like the disc is being read but it just stops trying (if you understand). Xbox one discs themselves work just fine but not after 360 discs.
I just checked out the disc troubleshooting and I will try out those fixes. I'll keep posted

Does the disc still work on the Xbox 360 console when you try to play it there?

Yes, the problem still exists. You know how when you put a game in, that it says the game name in the "home" area? It doesn't do that. It won't show the name until after I've finally got it to work by retrying the game over and over. When I try to play the
game from the HDD, it thinks the disc isn't in there. Sometimes it says the disc is unreadable. Most of the time it logs out of xbox live and logs back in. It even does this with new discs, and once it works, it works fine. This is driving me crazy.