Ok, I did a few checks with the multimeter. When the hazards are supposed to be on, there does appear to be a small voltage alternating between 0 and 0.1 volts reaching the light socket and the terminal block directly feeding the light socket. When the blinker is supposed to be on, there does appear to be a small voltage alternating between 0 and 0.5 volts reaching the light socket and the terminal block directly feeding the light socket. I get a negligible voltage when testing the voltage between the socket or terminal block and a ground point on the frame.
Just to make sure, I checked the blinker that still works and the socket and terminal is correctly alternating between 0 and 12 volts.
So could this be some sort of short in the wiring upstream of the light socket, or maybe a bad hazard switch or flasher module?