Most likely it is the sending unit. However, having taken mine apart, it is a bit difficult to explain how the low fuel light circuit could be closed if the float has lifted off the seat (needle is moving toward full). The sending units do get gunked up with diesel over time, and mine needed a good cleaning. This is an easy repair on the 123, but am less sure about the SD. You should be able to do a search on fuel sending unit in the Diesel Discussion and get some help.

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1976 240D
1987 560SL
2007 E320 Bluetec
1998 C280 (now son’s car)
1982 240D Manual - Sold