I suspect it is the stepper motor for the fuel gauge going bad. Low fuel light is usually pretty accurate since it is a very simple thing to turn on a light. What you describe is a pretty common problem with Avalanches, Silverados, and Blazers.
In this system, the PCM reads the resistance of the fuel sender in the tank and sends a digital signal to the instrument cluster on a single data bus (a wire) that all the signals for all the gauges are on. The instrument cluster has a processor that interprets this data signal and drives stepper motors for the gauges to where they are supposed to be. Other gauges working correctly indicates the PCM and processor in instrument cluster processor is working in general for the gauges. Typically a stepper motor goes bad. Ain't it wonderful!! If you think so, just study the electronics it takes to put your avalanche in 4WD with the push of a button.
What you want to do is read what the PCM says the fuel level is, using a capable scan tool. Car Gauge Pro for your Android phone and the BAFX Bluetooth OBD2 adapter (as sold on Amazon) will read the fuel level the PCM is seeing for about $30 total. Your gauge should match. If it doesn't and other gauges work fine, blame the stepper motor. If the PCM reading matches the gauge, and you know you have a certain fuel level - then suspect the fuel sender. Ebay has replacement stepper motors if you are tech savvy and handy with small fine work. Otherwise, replace the entire instrument cluster ($$$$$$) if a stepper motor is bad.
Good luck and best wishes for a successful repair.