Hi,
yes, you need a separate handle for publishing the screen capture, you cannot re-use the one you're using for publishing the webcam. Unless you're not actually publishing anything, and you will only alternatively publish video and screen, that is never send them at the same time: if that's the case you can re-use the same publishing handle, but you need to unpublish the previous stream first, and so create a new PeerConnection.
No need to create a new Janus object: the session remains one, you're just creating a new handle, something you already do anyway with viewers/subscribers in addition to the publisher.
L.