I now have the turn signal switch, and it must be the case that the small groove on the switch is worn out, and the switch is no longer held in place. I can push the switch in, but it pops back out.
Otherwise, all the lights and indicators work without problems. Since the switch is permanently pulled and the hazard lights don't work, I think the relay is also defective or something was removed by the previous owner. I'll get to work on it this weekend.