Look for burned out bulbs and dirty bulb contacts prior to replacing the flasher. Check the ground wires for the RH side turn signals, be sure to check front and rear turn signal wiring. The RH side front lighting uses a ground that connects on top of the radiator support and can easily be broken.
Note the black wire near the 2005 date on the following picture. It is the ground for the lighting on the RH side...
The fact that the flasher works OK for one side but not the other tells me that it is functional. The Hazard lights use a separate flasher from the turn signal flasher.
The flasher is not mounted on the front of the fuse panel. The flasher on the front of the fuse panel is a separate one for the hazard lights. It is in one of several places, depending on when your car was made. Some are behind the fuse panel on the back side, others are behind the radio and some are behind the glove box.
Other possible problem sources for the turn signal malfunction are the ignition switch, multifunction switch and the plastic shell that holds the turn signal wiring connector pins.
A word of caution about multifunction switches is in order here. The multifunction switch (high/low beam, wiper, turn signals) are different for different years. 87-98 will work in any 87-89 car. The 90-93 switches only work in 90-93 cars. You can't put an early model switch in a late model car, nor can you put a late model switch in an early car.
Supposedly you can move the pins around to make the switches work in model years that are different from the car the switch came out of. I cannot verify that and haven’t tried it .
The following diagram is for 87-89 model cars.
Turn signal switch wiring: