cannot login into iCloud

Recently I noticed that I can no longer send any texts via iMessage on Mac. I don't have an issue sending messages from iPhone but they consistently fail to be sent from the Mac. I found for similar issues and started trying to apply suggestions outlined in a similar report for El Captain. It didn't help.

In fact, once I signed out of my iCloud, I no longer can log back in. When I open iCloud settings and provide my credentials, it thinks for a bit and then cycles right back to the beginning of the wizard when the Next button is highlighted again as a first step to provide iCloud credentials. No errors, no nothing that would give any clue what's going on.

Also, I don't see the login key chain under available key chain list in Keychain Access utility.

What can I do to solve this issue?