A good example of multiple-room-handling for study may be tested right here on StackExchange, with the SE chat.
If you join multiple rooms, it simply has an extra list view which shows the names of the other rooms you are present in...and the last message posted in that room. There is a button in the list where you can switch your current view to swap to that room, and then your formerly current room becomes one in the "last message only shown" list.
I could take screenshots and describe it further. But the best thing to do is just go try it. If you want to post without bothering anyone, use the Sandbox:
Then try joining (for instance) the UX chat simultaneously:
Once you've joined multiple rooms open a browser window for each. Then look under the avatar list to see how it handles the list of rooms you are in with the latest message in that room.
If you're looking for an open-source clone of StackExchange chat, see Miaou:
It's MIT licensed, so you can
stealborrow without worries.
You might even want to implement your drawing app as an extension and avoid reimplementing all that functionality...because it allows you to integrate games in the chat. See Miaou Games where you can play "Tribo".
(Note: The developer of Miaou is (perhaps obviously) an SE user...and is likely willing to answer questions.)