Most GMs are of a design that the needle only moves under power. I was pretty sure yours was.
Ford, and most others will go to zero, or somewhere when power is removed. While GM's system eliminates the need for gauge voltage regulators, and Ford has adopted the same system, I have always considered it a flawed idea.
If you should lose power to the gauges, then they would stay where they were when the power was lost.
With no obvious indication that they were no longer giving real readings.
So, in your case, if you fill the tank and the gauge goes to full, then probably something has gone wrong with the sending unit in the tank.
If it does not move, then it still could be the sending unit, or the gauge may have failed or lost power.
I have driven a long time with a faulty gauge while waiting for a good time to drop a tank.
I usually know about what mileage my truck gets, and I reset the odometer every time I fill up.
I just drive until the number of miles I have driven are about the same as I would get out of 3/4 of a tank and then refill.
Doing that, will let you get by until you can determine what needs to be done.
If the tank needs to come out, I would throw in a 5 gal can, and drive until it runs out. Watching the mileage to be sure you are not likely to run out on top of a big overpass.
Then, pour in just enough to get you home. Thus the tank will be nice and empty when you go to drop it.
Just remember that an empty tank is much more dangerous than a full one when it comes to sparks.
Full tanks burn, empty tanks explode.