Welcome to TM Kim. The fuel level sensor is suspect if the problem is as severe as you suggest. The sensor is integrated with the fuel pump so maybe replacing this assembly might be prudent. I guess you could remove it as an interim measure and reconnect wiring such that with ignition on the effect of moving the sensors float arm could be monitored on the gauge. To avoid fuel pump running whilst doing this test the nearby 6 pin pump driver module could be temporarily unplugged.
