If all the hazard lights flash when the hazard switch is depressed then there is nothing wrong with the hazard switch, the wiring or the fuses.
The hazard switch is replaceable but more common are blown fuses, melted bulb holders or missing ground.
If only one side is not working when turn signal is operated on the Multifunction switch then the Multifunction switch is probably faulty....very remote chance that it is the Hazard Switch.
Download the 2003 KJ Service Manual here...includes wiring diagrams:
Index of /manuals/Jeep/KJ
It seems to me that there are not different fuses for Left or Right except for fuses 23 and 9 on the inside fuse panel when the Park Relay is activated.
If you look at the one page from the wiring diagrams I have loaded here...it shows the Mutifunction Switch at the top....this switches Ground G202 to either the Left Side or the Right side at the Hazard Switch.
So if you leave the Multifunction switch in the center position and touch a ground onto the Light Blue/White wire on Mutifunction switch pin C1 6 then the Left lamps should flash...touching Ground onto the Light Blue/Yellow wire should cause the Right lamps to flash. This will prove if the Multifunction switch is doing its job of switching ground over or needs replacing. Try Breaker yards if you really cannot find a new Multifunction Switch...there are two types depending on if you have front fog lights fitted or not. Export models are also different.
You can then follow the wiring out of the hazard switch on the Brown/Red wire for Right turn and on Dark Green/Red for Left turn to see if the +12 volts goes missing somewhere.