Gauge rises and falls from proper level to empty. It stay mostly in the low fuel range. It also registers in range mode what ever the gauge reads.

You replaced the fuel gauge stepper motor two times ? Did you do it correctly ?

Fuel Gage
The IPC displays the fuel level as determined by the PCM. The IPC receives a class 2 message from the PCM indicating the fuel level percent. The fuel gage defaults to empty if:
• The PCM detects a malfunction in the fuel level sensor circuit.
• The IPC detects a loss of class 2 communications with the PCM.
When the fuel level is less than a pre-determined value, the low fuel indicator illuminates.
Reviewing the fuel system description will further explain how the system functions. Refer to Fuel System Description .

Did you try replacing the fuel level sensor ? Check for DTC'S diagnostic trouble codes ?

Do you know anything about automotive electronic's ? The fuel level sending unit inside the gas tank is an input to the PCM/ECM / ECU - engine computer , the engine computer sends this information to the instrument clusters microprocessor over serial data network .The microprocessor controls stepper motor to control the gauge . Your best bet , take it to a qualified repair shop that has the proper diagnostic testing knowledge an tools , one being a professional type scan tool that can view fuel level sensor data to an from the engine computer .

I suggest you read the vehicle owner's handbook to be certain what the exact function of the light is. You will need to know that in order to determine whether the light is warning you or is broken.

The fuel light usually warns the fuel level in the tank is low...

The fuel level indicating system is a magnetic type system, which consists of the sending unit located in the fuel tank, an anti-slosh module located on the back of the instrument cluster, and a fuel gauge located in the instrument cluster. The sending unit changes resistance according to the level of fuel in the fuel tank, which varies the current flow through the gauge.The pointer position varies proportionately to the current flow. In this system, the sending unit resistance is low (22 ohms) when the fuel level is low and high( 145 ohms) when the fuel level is high. The pointer of the magnetic gauge remains in position when the ignition is turned to the Off position. On some models, an anti-slosh module is used to dampen out fluctuating fuel signals from the sender. Some vehicles are equipped with a low fuel warning indicator. The anti-slosh module will also actuate the low fuel indicator when the fuel level in the tank reaches 1/16 to 1/8 full.


Do your other gauges/warning indicators work properly with the ignition on? If not, check for power at the fuse panel/fuses.

If they do then partially remove your cluster with connector still installed and check for voltage at cluster connector and gauge. Check continuity of cluster and gauge for ground.

Sometimes the anti-slosh module (if installed) goes bad in the Grand Marquis but not likely.

Sender troubleshooting:

1. Perform test with resistors as follows:

1. Disconnect wiring connector at sender unit, then connect resistor between gauge lead and a suitable ground.
2. Turn ignition switch to On position.
3. With 145 ohm resistor, the gauge pointer should contact the Full mark.
4. With 22 ohm resistor, the gauge pointer should contact the Empty mark.

1. Perform test with fuel sender of known quality as follows:

1. Turn ignition to Off position.
2. Disconnect wiring connector from sender and connect it to test sender.
3. Move float rod away from fuel filter against Full stop position. Wait approximately 30 seconds, then turn ignition switch to On position. The gauge should read on or above the Full mark.
4. Move float rod toward fuel filter against Empty stop position. Turn ignition switch to Off position. Wait approximately 30 seconds, then turn ignition switch to On position. The gauge should read on or below the Empty mark. If gauge works properly replace sender, which involves dropping fuel tank.
   

The sender unit has a fine resistance wire with a slider to read the level, the slider or wires are worn and making intermittent contact, the low fuel light is a seperate switch so it will still work.access to the sender is usually under the rear seat cushon, under the soundproofing and a cover, have a look and describe what you have .

The gauge circuit is shunted on most cars, this is wierd, but needed to stabilize the readings... Your description sounds like normal function. Drivers would complain if the qauge were to be instant, going around corners, starting, stopping, all move fuel, and the gauge would indicate this, cauing driver alarm...Been this way for ever, although yours sounds like it is on the far side of the scale...if it reads correctly after a while, you could replace everything and you would have the same performance.

It is more likely to be the fuel guage sender which is faulty.

This is a normal concern.
The instrument cluster uses 4 different operating modes to calculate the fuel level:
Anti-slosh (default mode)
Key OFF fueling
Key ON fueling
Recovery
After a fuel fill up, the time for the fuel gauge to move from empty (E) to full (F) ranges from 2 seconds to 55 minutes depending on which operating mode the fuel gauge is in.
The default fuel gauge mode is called the anti-slosh mode. To prevent fuel gauge changes from fuel slosh (gauge instability due to changes in fuel sender readings caused by fuel moving around in the tank), the fuel gauge takes approximately 55 minutes to go from empty (E) to full (F).
The key OFF fueling mode (2 seconds to read empty [E] to full [F]) requires 3 conditions be met:
The key must be in the OFF position throughout the entire refueling of the vehicle.
At least 15% of the vehicle's fuel capacity must be added to the fuel tank.
The instrument cluster must receive a valid key ON fuel sender reading within 1 second of the key being put into the RUN position. The key ON sample readings are considered valid if the fuel sender reading is between 15 ohms ± 2 ohms and 160 ohms ± 4 ohms.
If these conditions are not met, the fuel gauge stays in the anti-slosh mode, which results in a slow to read full (F) event.
The key ON fueling mode (approximately 90 seconds to read empty [E] to full [F]) requires 3 conditions be met:
The transmission is in PARK (P).
The key is in the RUN position.
At least 15% of the vehicle's fuel capacity must be added to the fuel tank.
In key ON fueling mode, a 30-second timer activates after the transmission is put into the PARK (P) position (automatic transmissions). When the 30-second time has elapsed and at least 15% of the vehicle's fuel capacity has been added, the fuel gauge response time is 90 seconds to read from empty (E) to full (F). When the transmission is shifted out of PARK (P), the fuel gauge strategy reverts to the anti-slosh mode. The key on refueling mode prevents slow to read full events from happening if the customer refuels the vehicle with the key in the RUN position.
Recovery mode is incorporated into the instrument cluster strategy to recover from a missing fuel level input during a refueling event. Missing fuel level inputs result from intermittent opens in the fuel sender or its circuits. Recovery mode (empty [E] to full [F] approximately 20 minutes) is initiated when the following 2 conditions are met:
The instrument cluster is in the anti-slosh (default) mode.
The actual fuel level in the tank is greater than what is being displayed by the fuel gauge

your gas gauge sensor is most likley the problem on alot of new cars they measure your fuel off an average of were your float is at. like if your not moving it stays in one spot but as you move the float moves and reads the lowest amout that it is averaging and if your float keeps reading low your going to read that you have alot less gas than you really do. have you tried filling the tank all the way and see what it does? if it stays at full until you use some gas and your float starts moving your gas gauge will probley go down in that case its the fuel sensor if this dont help let me know and ill c what i can find out for you

You don;t say whether the fuel gauge is reading accurately, or whether it's always on empty. If the gauge is always showing empty, it's most likely a problem with the sending unit in the fuel tank. Check the resistance across the two terminals of the sending unit.

On a C900, the fuel sender level unit is separate from the fuel pump and located on the right side top of the tank, accessible by removing the trunk floor lift up panel (the pump is located on the left side top of the tank.
Possible problems are: electrical line off or broken, float not working (it's fairly delicate); or gauge not working.

Test the resistance of the sending unit with the lines disconnected, the resistance should be 2.6 to 3.8 ohms full tank, 63.3 to 67.5 ohms empty tank; and test for voltage across the top of the sender unit, if no voltage, that's likely the problem!

If the gauge is otherwise reading accurately (showing 'full' for a full tank, etc) then the problem is likely a gauge malfunction and can be simply replaced.

Good luck!

Saabreur