It has been 48 hours since I've reverted to my old ID (I was using a new ID as a work around). I have a couple of scenarios that usually replicate the problem. Most of my issues seem to be fixed such as: downloading/updating apps in the app store, links in email, and password settings for app store no longer disappear. But my Apple Music still has issues. When I try to play some songs, it takes multiple times of me pressing play for it to actually play. If I sign back into the work around ID that I created it works fine. Also I noticed that if I swipe closed the Apple Music app and then re-open it, it will give me an error occurred screen. It won't let me use any features on Apple Music. I have to sign in and out of Apple Music for it to work again. That problem actually happens on my original ID and my new ID. I just have to never swipe close the Apple Music app. Which after dealing with all the other issues for so long, I am perfectly happy with. Other than that everything seems to be OK.
I should also mention I am on iOS 9.2 beta 4. I have worked with the Apple Store Geniuses. They replaced my phone and once they do that, they pretty much no longer want to deal with you and just keep telling you to call Apple Phone Support. I've also worked with them and they had me send some logs to work with an engineer, but they ended up replacing my Apple Watch then I pretty much no longer heard from anyone after that.
I honestly no longer care to work on this issue anymore. I have a new phone, a new watch, and a simple fix that seems to work when problems happen. I am OK with that for now.