Bailey:

Try this: leave the harness unplugged from the WUR, start engine. Is there 12V continuous in the harness when the engine is running? If so, the relay and wiring are probably OK.

The lack of measured resistance of the WUR heater is puzzling; if it actually is zero the current should have been high enough to either damage the relay, or melt a fuse, or both. Does the ground connection in the harness show zero resistance to ground?

We did some more testing today. The harness does read 12 volts when the engine is running, and there is also continuity from the terminal to the ground of the battery.

This makes me wonder if we need to test the WUR again... The access to it is actually pretty terrible, and so it was hard to tell what was coming in contact with what.

Update! We took the G to mechanic who is well versed in old Beetles, and more importantly Jetronic fuel injection. They determined that the issue was dried up injector seals. At the same time they also adjusted the valves. The rich running issue has gone away, and the G is generally much happier. Fuel economy has also increased; from the low of 7 MPG up to 11 MPG.

I used to get a pretty consistent 15 mpg highway with my 84 280GE.

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1973 280SE 4.5 (Sold)
1984 LWB 280GE (Sold)
2000 G500 NMLE - "Deep Throat"
2007 ML320 CDI (Sold)
2010 Porsche Cayenne S Transsyberia TipTronic (Sold)
2010 Audi Q7 3.0 TDI (Sold and Sorely Missed)
2014 ML350 BTC
2015 Audi Q7 3.0 TDI (On Order)