Does that switch also control the cruise control, the headlights and the headlight dimmer on that model? If so, it sounds to me like you might have to replace the whole switch. There's an earlier thread here that describes that job. It's not too bad a job. The last switch I replaced cost me about $60 US.
make sure it is not just a burned out bulb. blinker bulbs have 2 fillaments, one for blinking and another for running lights. GL
The 4 way and the turn signals have 2 separate flashers. Try replacing the turn signal flasher.
Easier to just jack the car up so that you can get under it look for the blinker bulb assembly twist it should come out if you line up the holes replace and reinsert
My Haynes manual says that the "96 turn signal/hazard flasher is plugged into the "convenience center" which is mounted at the top of the left kick panel. FriPilot
Behind the front right blinker
I just had to change out the front blinker bulb on my 97 sunfire gt, and as far as I know, there were no huge revisions between the 97 and 98 model year, so here's what you'll need to do. First of all, take out the headlight above the blinker. Your manual should tell you how to do this. Then, simply reach through the hole you've just created to losen the nuts holding the light bulb casing in place, pull it out, and change the bulb. Simple!
try replacing the flasher. there should be two, one for flashers and one for turn signals
the plastic cover over the head light has a bunch of plastic retaining clips which must be removed by pulling the center up then there is two torx bolts which hold the head light on
Check in the driver's side footwell in the side kick panel. There might be something called The Convenience Center which houses those little guys. If your doesn't have the "center", check inside the steering column housing. You might try turning the flashers or a blinker on and listening to hear where the clicking comes from.
Is your headlight switch and blinker switch the same? If so, this is called the combination switch. Sounds like it is bad and needs to be replaced. Not uncommon for these to break.
Turn on the hazard switch (emergency blinkers). You should hear it under the dashboard on the drivers side. The emergency blinker and turn signal are each controlled by a different relay but the relays look alike. Once you've located the emergency blinker relay, the turn signal relay is nearby.
Go to Autozone. When you purchase the bulbs they will put them in for free.
Look on or around the fuse panel under the dash on the drivers side! It is round!
You might need to refill your blinker fluid. If that does not work, check above the passengers visor..