Markulta,
Pull the connector at the top of the sending unit and drive for a while to see if the reserve light goes on. If it does you have a short in the wire. If it doesn't then the problem is almost definitely in the sending unit. The contacts are at the bottom (you'll have to remove the tube cover) and the circuit is normally closed when the float sinks all the way down. Something is likely shorting it.

Dcraig,
I agree that the sudden fluctuations indicate the fuel is leaking out of the tube way too fast. I'd take it apart and try to see where it might be leaking. The drain at the bottom is actually a kind of serpetine maze that the fuel needs to run through. Also, there should be one or two small (pencil lead sized) holes in the side of the tube. If these have been somehow enlarged that could be the problem. Also, you may want to clean the wire contacts that the float runs up and down on. I found that mine were really quite dirty and the contact was being compromised so I was getting an eratic reading. I used mineral spirits and a Qtip, being really careful not to let any of the cotton threads from the Qtip to get stuck in the mechanism. The wires look pretty fragile but I found if I was gentle it was no problem cleaning them.

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LRG
1987 300D Turbo 175K
2006 Toyota Prius, efficent but no soul
1985 300 TDT(130K miles of trouble free motoring)now sold