I'm not familiar with MoHo fuel gauges, but I've got lots of experience with Classic cars.
Since the gauge moves slightly when you turn on the key there is power to the gauge and sending unit, but one or both are not working.
A common problem with fuel gauge sending units is poor connections. This is particularly problematic for vehicle which sit a lot, as the fuel turns to varnish it leave a sticky coating on the sending unit which interrupts the signal. I've read online on Mustang forums and personally found that Chevron Techron fuel treatment works well. 05-07 Mustangs have this problem and a periodic use of techron seems to keep my car's gauge working.
The float may have sunk, in which case you would need to replace it. If you remove the sending unit, you can test the float for buoyance in a pail - be very careful, fuel is flammable!
1977 Safari Land Yacht
2005 Toyota Tundra SR5
2010 Ford Flex Ecoboost