I also have a 2013 378XL. I have run the generator all night and have not experienced and issues with exhaust or fumes. Since we have an all electric refrigerator, unless you are plugged in or engine running, I would hate to wake up with dead batteries or food starting to thaw. I was worried about generator noise when sleeping but it wasn't bad.
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Mickey & Sue Stamper
2013 GT XL 378TS
2013 Ford Escape on Landgrebe tow dolly