gas gage

Many GM gas gauges are notorious for this, especially if the owner(s) have used lesser grade fuels. The windings on the sensor will often corrode and as a result, there will be no contact with the wiper. Not sure about the STS but often this requires replacing the entire fuel pump as they are part of a single assembly. There are some additives (available from GM) that will sometimes correct this but they are usually a limited time cure.

There are of course other possibilities for this problem as well.

EChas3 said:

The system is designed to provide a stable reading. When it's near empty, fill it up and see if it works properly.

If so, just don't get freaked-out when a small purchase doesn't register right away. It will in a while.

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This should not be the case if you turn off the vehicle during the fill. This doesn't apply once the ignition is cycled. The PCM will perform a new assessment on every ignition cycle.

Is it safe to assume the gas gauge sweeps the same as the others on the cluster when you first start your car?

The problem then will be either with the sender (most likely) or one of the following:

Wiring/connector from the fuel tank to the PCM.
The PCM itself (least likely).

The fuel level is sent as a serial data message from the PCM to the cluster. If there was a data bus problem to your cluster, you would notice virtually every other gauge/function would not work either.

$20.00 is a fair amount of fuel. Here, that will buy me about 1/3 tank (of premium) so there should be no problem generating gauge deflection under normal circumstances.