Operators are going paperless now, so signing and leaving a copy of the loadsheet are outdated. We sign our loadsheet(and accept the flighplan) electronically these days via ACARS.
That said, I would not depart until all the "paperwork" is settled....no need to take problems with you into aa complex operating environment. It's bad enough having to tell ATC that we are waiting for the cabin - I wouldn't want to tell them that we are clogging up their airport waiting for paperwork that should have been sorted ages ago...