I will bet one dozen hot Krispy Kreme donuts that your sending unit is locked up and there is a lot of gunk and rust inside the tank.
I would pull the tank out where you can work on the screws easier, and have better light so you can see what's inside the tank. A radiator shop can clean out the gunk, but you might be money ahead by just buying a new tank. They are not all that expensive. Under $200.00 including shipping. A new tank is a wonderful source for peace of mind. No worrying about varnish migrating to the internals of everything mechanical and gumming up the works.
