We're also having issues with the shoutbox on our community. It get stuck a lot, and on mobile the textbox sometimes does not show for some reason. I've found that if you are inside of a topic or personal message (instead of the homepage), the shoutbox gets sticky and won't close.
I have enabled the full-screen chat while still keeping the shoutbox active. I asked the community members to use both and see which they prefer. The full-screen is a clear win at the moment. The only downside is that you don't have an icon in the notification bar like you do with the shoutbox. As a workaround, I've edited the CSS as @fearlessfrog suggested and added a link to my chat category.
I think a great improvement to the full screen category chat would be a way to configure an icon in the header that also has the unread messages count for that chat like the shoutbox has.