Blinkers only blink right at first, after a second they just stay lit and don't blink.
Check the blinker relay.
That sounds like the loads are wrong, check all of your turn signal lamps If you even have one lamp out on that circuit it can throw the loads off, if all of the lamps are good then check lighting grounds, or it could be a bad flasher.
I agree with tennisshoes, It most likely will be one bulb out. If the left light is the issue, go to the other end of the vehicle and see if the light is working.
Looking for a Used Cavalier in your area?
CarGurus has 1,376 nationwide Cavalier listings starting at $1,000.
Search Chevrolet Cavalier Questions
Chevrolet Cavalier Experts