If you are able to start the engine with the boost switch, then the path from the chassis battery to the engine is OK, and so is the engine starter system. The symptoms really do point to a discharged ("dead") chassis battery.
If you're looking for the starter relay, you may be able to find it by looking under the hood and through the wheelwells. That would save you the trouble of opening and closing the doghouse.
Although I have a service manual for our model-year F-53, I leave figuring out engine and drivetrain problems to our trusted local mechanic.
2008 Holiday Rambler Admiral 30PDD (Ford F-53 chassis)
2009 Honda Fit Sport