iMessage Apple ID. verification failure

I cannot use my message app ever since I got the prompt window "Sign in wit) your Apple ID to activate iMessages. When I enter the Apple ID and password (which I have changed once before and which I have tested in iCloud account to make sure that it works, and it does), the window tries to verify the Apple ID, but after long time of loading, it comes back with a ,estate that verification failed, or cannot connect to servers. After, the message list is frozen.

This window prompt pops out every time I start the messaging app and it freezes it.

What can I do? I don't even want messages but I need to read a lot of received messages I have received and haven't had a chance to read yet.