One of the wires going to that solenoid is a constant 12V+ and the other is a negative ("ground") that is controlled by the PCM. When the PCM wants to open that valve to let the engine suck in the fuel vapors, it connects that wire to a ground which completes the circuit so power can flow which activates the solenoid. It's just like having your tongue always on the positive terminal of a 9-volt battery: you will never feel anything because the power isn't flowing. As soon as you rotate the battery so that the negative touches your tongue, the power flows and your tongue gets zapped. The PCM grounded the EVAP circuit to open the solenoid, but saw that the circuit was still open, so it threw a circuit code. Or maybe it saw current flow (closed circuit as commanded), but it saw TOO much flow, indicating a short in the circuit (shorted solenoid).
With key on, and negative lead of your DVOM on a known good ground, you should have 12V+ at one terminal. Switch your DVOM to test continuity and probe the other terminal and you should not have continuity unless the PCM is commanding an EVAP purge (and it shouldn't be). The solenoid is easy to remove. Pop it off and try to gently blow through it. If you can, then it's stuck open and needs replacement.
I still don't see how this would have any effect on fuel gauge accuracy or fuel economy.
UNLESS... The EVAP system has a leak somewhere and the fuel is dripping out or evaporating. You're not getting worse MPG, you're just losing gas. And maybe this leak is allowing air to get in the tank which introduces moisture for the ethanol to absorb which causes the rheostat of the level sensor to corrode and read incorrectly...
That last paragraph was really farfetched. Besides, that would throw something like a code P0455.