When I open up the fuse box near the engine, I hear no click when I turn on the regular headlights (low beams), yet I do hear a click when I turn on the high beams. I tried to switch what I thought to be the relays for low and high beams, but no improvement - with the caveat that I'm a little confused about which are the relevant relays. I do not believe this is a bulbs issue since this happened with both bulbs at the same time. Any counsel?
Most of the time it's just the headlight assembly fogged up and needs polishing with a kit. Halogen low beams when new were not super bright and they are aimed low to avoid blinding other drivers. You may be used to low beams on newer cars that are led. You need to check the voltage at the headlight connector under load. Meter plus and minus at the connector, not using chassis for the negative lead. The relay is part of a long multiple relay module called the integration relay. It is unlikely to be the issue. Each low beam has its own ground point so the ground point itself is unlikely. You could try a led replacement, which are technically illegal.
These led versions were recommended by one of the more reliable members of Priuschat. Again they are technically illegal and a fogged up headlight assembly will still be an issue. The leds get very hot at the mounting and this low cost version is one of the first without a built in fan.