Using the procedure above, you will probably discover that the poorly-engineered GM sender unit is the issue. It's very fragile - a terrible design, probably by the same engineer that did the dew wipes.
I bought a "universal" sender on eBay for $22 including shipping, and discarded everything but the swinging arm and the base holding the resistor coil. That base mounted nicely on the original support with nylon (fuel-proof) cable ties, and I bent the arm so it would have the same swing as the original. Resistance is 90 ohms full, close to zero empty. Works perfectly and is very stable. It's a robust solution that might outlive the rest of the car.
The biggest task is R&R of the fuel tank. While it's out (and empty), you might remove all the crap that will have accumulated, and have the tank cleaned by a radiator repair shop. Don't be surprised if an internal baffle has cracked or broken (just remove the broken pieces - it'll work ok). Fuel pumps are cheap compared to the work involved, so if you still have the original it might be a good idea to replace it and the screen.
One other point. The wiring to the fuel pump might need freshening to ensure that there are no high-resistance connectors in the circuit. Just pulling, cleaning, and replacing the connections will sometimes make a measurable difference in the voltage detected at the pump (when operating).
Before reassembling, take a close look at the rubber lines that connect the metal lines from the fuel tank to the car.
And while the tank is down, you have a chance to apply rust reformer or other treatment to the cavity it normally blocks.
[This message has been edited by notaguru (edited 04-04-2014).]