Instrument panel warning signals

Turn Signal Lamps
Ignition positive voltage from the HAZT/SIG fuse 15 in the instrument panel fuse block is supplied to the hazard warning switch. The turn signal lamps may only be activated with the ignition switch in the ON or ACCY position. When the turn signal switch is placed in either the left or right position, the flasher module sends an ON/OFF voltage to the turn signal switch assembly. For the rear turn lamps, battery voltage is applied directly to either the left or right turn signal lamps. The ground for the rear turn lamps is supplied from G401. For the front turn lamps, battery positive voltage is applied from the turn signal switch to the instrument panel electrical center. From the instrument panel fuse block, battery positive voltage is applied to the instrument cluster (IC) indicators and to the normally-open switched side of either the DRL L or DRL R relay, depending on the turn signal switch position. Battery positive voltage is then applied to either the left or right turn signal lamps. The front turn signals are grounded at G102.
For the hazard lamps, voltage is applied from the STP HAZ fuse 8 to the hazard warning switch. When the hazard switch is pressed, all the turn signal lamps will flash, including both IPC turn signal indicators. The hazard warning switch is grounded at G201.


Your best bet , take it to a qualified repair shop.

What is it chiming ? Chimes for different reasons , headlamps on , seat belt not buckled , etc.... What warning light is lit on the instrument cluster ? if any . seat belt , air bag's etc...

Audible Warnings Description and Operation
The audible warnings alert the driver of a system concern or a critical vehicle condition. The following lists the audible warning priority and the pulse rate:
Fast rate chime (200 pulses per minute)
Medium rate chime (150 pulses per minute)
Slow rate chime (50 pulses per minute)
Single chime
Fasten Safety Belt Warning
The inflatable restraint sensing and diagnostic module (SDM) activates the fasten safety belt audible warning for 5-8 seconds. The fasten safety belt warning sounds and the fasten seat belt indicator illuminates when the following occurs:
• The ignition switch is in RUN, BULB TEST, or START position.
• The SDM detects that the driver's seat belt is not buckled (signal is low).
If the seat belt is buckled before the ignition is in RUN, BULB TEST, or START position, the chime does not sound. If the seat belt is buckled while the chime is sounding, the chime stops.
Key-In-Ignition Warning
The instrument panel cluster (IPC) activates the key-in-ignition audible warning. The key-in-ignition warning sounds when the following occurs:
• The ignition switch is in OFF, LOCK, or ACCY position.
• The body control module (BCM) determines that the driver's door is open (signal is low).
• The BCM determines that the key-in-ignition switch is closed (signal is low).
Lights On Warning
The instrument panel cluster (IPC) activates the lights on audible warning. The lights on warning sounds when the following occurs:
• The key is not in the ignition.
• The body control module (BCM) determines that the drivers door is open (signal is low).
• The BCM determines that the headlamp switch is in the park or head position.
Rear Parking Assist
The rear object sensor control module activates the audible warning. The audible warning sounds whenever any of the indicators are illuminated on the rear parking assist display. For a description of the rear parking assist system refer to Object Detection Description and Operation.
Turn Signal Reminder
The instrument panel cluster (IPC) sounds the turn signal reminder once per each flash of the turn signal lamps when:
• The ignition switch is in the RUN position.
• The turn signal switch is in either turn position, sensing, within 1 second, an open to B+ transition in the turn signal control circuit (left or right turn).
• The vehicle has traveled a distance of 1.2 km (0.75 mi) by counting pulses on the vehicle speed input circuit.
The IPC turns off the turn signal reminder when either the ignition switch is turned to the LOCK position or to the OFF position, or the turn signal switch returns to the OFF position, sensing, within 1 second, no transitions from open to B+ in either of the turn signal control circuits (left turn or right turn).
Additional Chimes
The following chimes will turn on when an associated indicator turns on in the instrument panel cluster (IPC):
• Low Fuel Indicator
• Low Oil Pressure Indicator
• OVER SPEED indicator
For a description of an indicator refer to Indicator/Warning Message Description and Operation.
For diagnosis of an indicator refer to Symptoms - Instrument Panel, Gages and Console.

Your have a problem with one of these systems , your best bet , take it to a qualified repair shop. They would do diagnostic's to find the problem .

Do any indicators illuminate when the chime sounds?
Go to Symptoms - Instrument Panel, Gages and Console
Go to Step 3
3
Turn the ignition OFF.
Remove the key from the ignition.
Turn the headlamp switch to OFF.
Close all of the vehicle doors.
Open one or more of the vehicle doors
Does the chime sound at all times regardless if the doors are open or closed?
Go to Step 15
Go to Step 4
4
Close all of the vehicle doors.
With a scan tool, observe the LF Door Ajar Switch parameter and the RF Door Ajar Switch parameter in the instrument panel cluster BCM/Inputs data list.
Do all of the Door Ajar Switch parameters display Door Closed?

With a scan tool, observe the Left Rear Door Ajar Sw. parameter and the Right Rear Door Sw. parameter in the instrument panel cluster BCM/Inputs data list.
Do all of the Rear Door Switch parameters display Closed?

This is a 21 step diagnostic trouble shooting flow chart .

Important: The hazard warning switch includes the turn signal and hazard warning flashers.
Removal Procedure
Remove the center air deflector. Refer to Instrument Panel Center Air Outlet Deflector Replacement .
Squeeze the retainer on the backside of hazard warning and cruise control switch.
Pull the hazard warning and cruise control switch from the retainer in the instrument panel.
Disconnect the electrical connectors from the hazard warning and cruise control switch.
Remove the hazard warning and cruise control switch from instrument panel.
Remove the hazard warning and cruise control switch bezel from the instrument panel trim pad by using a small flat-bladed tool to the release retainers.
Remove the radio. Refer to Radio Replacement in Entertainment.
Reach through the radio opening and push the retainers up towards the roof of the vehicle. When the retainers release, slide the instrument panel (I/P) center air outlet deflector out of the I/P opening
Remove the air outlet deflector
Remove the instrument panel accessory trim plate. Refer to Instrument Panel Accessory Trim Plate Replacement in Instrument Panel, Gauges, and Console.
Remove the bolts from the radio.
Pull the radio toward you.
Remove the front floor console trim plate. Refer to Front Floor Console Trim Plate Replacement .
Pull the instrument panel (IP) accessory trim plate away from the IP trim pad.
Disconnect the IP harness seat heater leads, if equipped.
Remove the seat heater switches from the IP accessory trim plate, if equipped.
Remove the IP accessory trim plate.

There is no relay , controlled by integrated power module . Electronics !

YOU still have a short or ground in the wiring most likely in the runnign light circuit.

Symptoms point to the cluster microprocessor. How was it proven? See diagram below.

When signal lights stay on at indicators there's usually bad bulbs or the headlight switch is no good it's a self warning in the system check the bulbs for good elements if okay I'm pretty sure that the headlight switch is n / g

nope you got a bad ground either at the light sockets or body ground

check the multi function switch assembly (ie..turn signal stock) comes as a whole unit.

Check out this recall below:
Bulletin No.: 04098A

SPECIAL POLICY
Subject:
04098 - Special Policy Adjustment - Turn Signal/Hazard Warning Flashers 

Condition
Some customers of 1999-2002 Chevrolet Malibu, Oldsmobile Alero, Pontiac Grand Am, and 1999 Oldsmobile Cutlass may have turn signal/hazard flashers that become inoperative.

Special Policy Adjustment
This special policy covers the condition described above for a period of 10 years or 150,000 miles (240,000 km), whichever occurs first, from the date the vehicle was originally placed in service, regardless of ownership. The repairs will be made at no charge to the customer.

Good luck and hope this helps:
HAZARD WARNING SWITCH REPLACEMENT - ON VEHICLE 
REMOVAL PROCEDURE 

Remove the center air louver deflector. Refer to Air Deflector Louver Replacement - Center Vent. 
Remove the instrument panel cluster. 
Remove the hazard warning switch bezel from the instrument panel using a small-flat bladed tool to release the retainers.
Remove the left side bolt from the hazard warning switch through the instrument panel cluster pocket Remove the right side and center lower bolts from the hazard warning switch through the center air louver deflector pocket.
Reposition the hazard warning switch from the instrument panel to access the electrical connector. 
Disconnect the electrical connectors from the hazard warning switch. 
Remove the hazard warning switch from instrument panel. 

INSTALLATION PROCEDURE
Connect the electrical connectors to the hazard warning switch.
Position the hazard warning switch to the mounting points in the instrument panel. NOTE: Refer to Fastener Notice in Service Precautions.
Install the left side bolt to the hazard warning switch through the instrument panel cluster pocket. Tighten Tighten the left side hazard warning switch bolt to 3 N.m (27 lb in) . 
Install the right side and center lower bolts to the hazard warning switch through the center air louver deflector pocket. Tighten Tighten the left side hazard warning switch bolt to 3 N.m (27 lb in) . 
Install the hazard warning switch bezel to the instrument panel pressing into place until fully seated. Install the instrument panel cluster. 
Install the center air louver deflector. Refer to Air Deflector Louver Replacement - Center Vent.