Sooo, what would make my gas gauge stop working after filling the tank the other day? It is staying on full. Has never happened before.
take a look at this discussion, lots of things discussed.
The float might be stuck.
Take a rubber mallet and give the gas tank a few good whacks to see if it gets the gas gauge to work again.
Tester
Could be a couple of things:
-Stuck float (do what @Tester suggested)
-bad dash gauge (does it move when you turn the key on? or does it just always say full, no matter if key is on or not?)
-someone is messing with you and topping off your gas tank each night after you go to bed.
If it is #3- send them my way 
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Tester, I will give that a try. Thanks
eddo, even when the gauge was working, it would stay at the level of the fuel in the tank even when the ignition was off.