Check for power at the buzzer with the key in the ignition and the door open. Does your dome light work with the door open? If it does, then you know the door switch is good. Could be you have a bad buzzer.
I'm looking at the factory manual for my Cutlass right now. It says you could have a bad turn signal switch. You need to check the buzzer continuity at the connector on the steering column. If you look at the column from underneath you will see a flat connector with multiple wires. You need to take it apart and use a paper clip (bent in a U shape) to check the connection at the column. Without a service manual it would be hard to explain it. It's not hard to figure out but without the pics I have you would only be guessing. It also says you could have a bad key lock cylinder of all things. Find a factory manual is my best answer.
God luck.