Unable to Log in to game account; game client does not shut after I quit game.

So I log on to my account from two different devices: a desktop, and a laptop.

Currently my laptop attempts to log into game account, but says "Server is Unavailable"; then gives me the option to play offline.  (Keep in mind, when I attempt to play game on my desktop, my desktop logs into account just fine.)

I then quit game, but then on my steam library it shows the game still in "running" state.

This only started happening after I chose the option to "play offline" one time.  Every time before that I could successfully log in.

Please assist in resolving  Client log file given below.