Gas gauge troubleshooting

if the water is cold and the temp gauge needle moves at all at switch on... something is wrong. It could be the temp sender but my cases they simply quit working, no needle movement at all.
There is a white plastic engine harness connector under the hood. these will melt down and short wires together. The purple temp sender wire is in this connector.
Wild needle swings is a sign of faulty mechanical limiter. Most often seen at switch on with pre heated water. The added ground wire you describe doesn't always work because the fuel sender is pretty much isolated by its mounting gasket. If the lock ring at sender and the tank hanging straps were a guaranteed low resistance chassis ground path the factory would not have added the ground jumper on the sender.
The possibility exists that a faulty limiter has already killed the fuel gauge. So like others have said, look at other parts of the circuits first, maybe end up in the instrument panel.