I have the same problem as well with El Capitan, but it's not the first time. I've had the "cannot log into server" error before. I had it with Snow Leopard. Mavericks didn't have it. I skipped the Yosemite upgrade. Now it's back again with El Capitan, so this is NOT a new problem. Couldn't sign in with iTunes OR Apps Store. Furthermore, I didn't always have it with El Cap. When I first bought my MacBook Pro it was fine. But at some point, not sure when, I lost the ability to log into iCloud.
TEMPORARY SOLUTION:
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You said that you cannot login to your AppleID, unless you are in Safe boot mode.
Can you log in to your Apple ID when you are logged in to another account on your Mac?
Make a new user just for troubleshooting this issue. You can do that in your Users & Groups pref pane.
Be sure that the new account is an admin account (check the box: Allow user to administer this computer)
Log out of your present account and log in to the new user. Quickest step then would be to log in to either iTunes, or the App Store. You would have to login successfully to View your Account.
I like iTunes for this test, as the account quickly lets you see how many devices and computers that you have authorized.
You may see some kind of difference between "devices" and "computers", so look at that. You may find out that you still have some old devices that are still authorized, but that you no longer possess. You can even "deauthorize" all devices, then re-add the ones that you are certain about. Another choice would be to make sure that you are logged OUT of your AppleID, then log back in successfully a second time.
Or, maybe you just need to do some more checking on other software that you have installed that may be affecting your login to your AppleID.
When was the last time you checked for malware on your Mac? The software
Malwarebytes is quite good that that, and only takes a few seconds to run.