Voltage increases when the engine is running because you're now getting "charging" voltage from the stator. Same reason why voltage increases when a car engine is running because now the alternator is working. That part is normal - in other words, it "IS" battery voltage since the battery is receiving 14+ charging volts.
Go to the access for your fuel tank level sender. You will see a black (possibly green, but same function) wire and a pink wire. The pink wire is an industry standard for the level sender. You may see that it's not connected/rusted/corroded/dirty... if so, fix that first. Next, with the key to the 'on' position (gauges activated), you can do two things. Ground the pink wire... the gauge should show full. Remove the pink wire, the gauge should show empty. You can do the same type of test at the gauge to figure out where the issue is. If you have 12V at the ignition post on the gauge, yet when testing the gauge the needle doesn't move, then the gauge is bad.
NOTE... the pink wire carries resistance. You should have a separate terminal on the gauge that receives ignition ("run" position) voltage to power the gauge.
NOTE 2: I see that you are measuring 11V in one place and 12.5V in another (with engine off). The 12.5V is good. The 11 is not. I suggest you start with pulling/inspecting/cleaning/tightening every connection you have back there. If you're not getting clean voltage everywhere, then the everything else you are doing may be a waste of time.
Note 3: It seems like you're kinda all over the place with understanding terminology, function and proper wire colors... which is fine... and understandable if you're new to this. But take a step back and go through things again to fully understand. If you're uncomfortable with this stuff, just hire someone. A decent shop can have this figured out in, likely, less than an hour.
EDIT: Since you have a combination voltage/fuel gauge, the power post is likely shared between the two functions.
Go to the access for your fuel tank level sender. You will see a black (possibly green, but same function) wire and a pink wire. The pink wire is an industry standard for the level sender. You may see that it's not connected/rusted/corroded/dirty... if so, fix that first. Next, with the key to the 'on' position (gauges activated), you can do two things. Ground the pink wire... the gauge should show full. Remove the pink wire, the gauge should show empty. You can do the same type of test at the gauge to figure out where the issue is. If you have 12V at the ignition post on the gauge, yet when testing the gauge the needle doesn't move, then the gauge is bad.
NOTE... the pink wire carries resistance. You should have a separate terminal on the gauge that receives ignition ("run" position) voltage to power the gauge.
NOTE 2: I see that you are measuring 11V in one place and 12.5V in another (with engine off). The 12.5V is good. The 11 is not. I suggest you start with pulling/inspecting/cleaning/tightening every connection you have back there. If you're not getting clean voltage everywhere, then the everything else you are doing may be a waste of time.
Note 3: It seems like you're kinda all over the place with understanding terminology, function and proper wire colors... which is fine... and understandable if you're new to this. But take a step back and go through things again to fully understand. If you're uncomfortable with this stuff, just hire someone. A decent shop can have this figured out in, likely, less than an hour.
EDIT: Since you have a combination voltage/fuel gauge, the power post is likely shared between the two functions.
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