Screwy Gas Gauge

['68OrangeSunshine]

The Fuel level indiccator on my '03 Chevy C/1500 Tahoe has been acting weird lately. Sometimes, it will stay on Empty and the yellow light stays on, even after topping off the tank.
The other night it was sometimes showing about 9/16ths full, then dropping to half full, then almost zero.
I filled it up. Tank took 13.8 gallons. Legitimately a half tank, but the gauge stayed on empty and the yellow idiot light stayed on.
Once up in the Foothills, after midnight -- and no cops around -- I made some quick, G-inducing lane changes -- to slosh the tank internally. It worked for about a minute, then the needle sank again. It's been doing this for a couple months.
Then sometimes, the needle pops back up and reveals the true level.
IDK.
I have been logging trip miles, fill-ups and gallons taken onboard. My MPG has gone down.

Where do I start troubleshootig?