You have two separate systems, and two separate fuses for the turn signals and the hazard flashers.
Turn signals are fused on one fuse through an ignition feed.
Hazards are fused on another fuse through a battery feed.
Those two fuses feed the flasher, to the multifunction switch.
One to the turn signals (ignition powered fuse) the other (battery powered fuse) to the hazards.
Why? So that the hazards work with the ignition off, for safety reasons.
Anyhow, if all is working with all the lamps except the hazards, look for a short on the battery feed fused lead, between the switch and flasher.
It sounds like when the switch is turned on it's completing a circuit to a dead short.
Thus blowing out the flasher and fuse.
97 Sahara, 74 Renegade, 72 CJ5, 55 Willy's (the Beast) Boss 302, 6:34 gearing, Bose 901 speakers.
God I miss that Jeep. Forgot the kinda Jeep, 80 SX4.