As advised, use a meter to determine which bank os being charged from shore power and engine alternator. And yes, you can simply disconnect one bank and then see if the engine starts from the other one (assuming you battery parallal solenoid is not stuck closed).
Engine alternator only charges engine batteries. Shore power only charges house batteries. That was one thing that bugged me when I first bought my coach, unless the dashboard battery parallel button is held in. I think Winnebago changed that arrangement on later model year coaches, but my 02, and I think your 03, are the same.
I replaced my parallel solenoid with a Blue Seas smart switch, so that low batteries first, then all batteries, are automatically charged from any source, whether it be shore power, engine alt, or generator.
BTW, Skip's suggestion is excellent.
USAF 1965-1971; USCG Master
2002 Horizon 36LD - 2000 GMC Sierra 1500 4x4
Indian River, De