Although most of the circuitry, including the Hazard switch, is common to both the turn signals and the hazard signals, there are separate fuses for each. If the turn signals work correctly, this would isolate the problem to what's unique to the Hazard function and that includes the separate fuse. The fuse for the Hazard lights is in the in-cabin fuse panel (15 Amps, possibly #22). It also powers the AT shift lock control module, key warning switch, and the power antenna, so if the fuse is blown, these also might not be working.
If the fuse is blown, replace it with the same size fuse only. (Although fuses sometimes fail due to aging, more often than not they fail because of a problem in the circuit they're meant to protect. Consequently, the replacement should not be greater. If the replacement fuse blows, there's a short in the circuit and more troubleshooting will be needed.)