I also have this problem, where Facebook keeps signing me back into chat after I go offline.

I found a reasonable, though not perfect, workaround:

Go to the "advanced settings" of the chat sidebar, choose the option "Only some friends see you...", and fill in one name only. Choose a person that you're OK with him/her seeing when you're online (I chose my wife :), and that way the rest of your Facebook friends will never see your online status, and you can remain signed in to chat.

I also have this problem, where Facebook keeps signing me back into chat after I go offline.

I found a reasonable, though not perfect, workaround:

Go to the "advanced settings" of the chat sidebar, choose the option "Only some friends see you...", and fill in one name only. Choose a person that you're OK with him/her seeing when you're online (I chose my wife :), and that way the rest of your Facebook friends will never see your online status, and you can remain signed in to chat.

I also have this problem, where Facebook keeps singing me back into chat after I go offline.

I found a reasonable, though not perfect, workaround:
Go to the "advanced settings" of the chat sidebar, choose the option "Only some friends see you...", and fill in one name only. Choose a person that you're OK with him/her seeing when you're online (I chose my wife :), and that way the rest of your Facebook friends will never see your online status, and you can remain signed in to chat.

I also have this problem, where Facebook keeps signing me back into chat after I go offline.

I found a reasonable, though not perfect, workaround: 

Go to the "advanced settings" of the chat sidebar, choose the option "Only some friends see you...", and fill in one name only. Choose a person that you're OK with him/her seeing when you're online (I chose my wife :), and that way the rest of your Facebook friends will never see your online status, and you can remain signed in to chat.