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This almost sounds like a poor wiring connection. I would first inspect the fuel sending unit connector near the fuel tank. I would make sure this connector is not corroded and has a good connection, you may even want to get some electrical tape and and put some on after you plug the connector back in. Wiring issues like this can be difficult to troubleshoot. I would try to follow the wiring harness the best you can and look for any spots in my be rubbing or pinched. Normally a bad fuel sending unit will read full all the time. Below is a picture of the connector for the fuel sending unit you are looking for. I hope this helps.
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