Why does fuel guage go from half full to...

Intermittent connection, might be the senders chassis ground path.
It might be on the positive side too, like the speedy nuts that mount the gauge in and to the printed circuit board.
You didn't say what model. So I didn't reply.
The gauges have threaded studs mounted in a piece of fiber board on the back of the gauge. These fiber boards warp with age. Nuts get loose. Thermal gauges have temperature changes, some a lot more than others. So cold makes contact, then heat/movement breaks the contact.
I wont go into the 3 post fuel gauges also having a ground where they mount in the inst' panel, just ad that these are where the most temperature change is, where the more common loose connections are.
Good luck with it