My cavalier's daytime running lights stay on whether the switch is on in the interior or not. The headlights and high beams do not work. We've checked the bulbs and they work, checked the wire harness and that works. Would it be the relay? or the switch next to the steering wheel?

It could be either. You'll need to check if the signal is getting to the relay to be able to narrow it down. Or, you could just buy a new relay and try it. The relay is much cheaper and a helluva lot easier to install.