It sorta sounds as if your Inverter/converter is faulty. The lights are 12VDC, so being hooked up to shore power shouldn't make a difference....other than that it's charging your batteries and the converter is providing the 12VDC.
I'm guessing that you've got bad batteries....maybe a dead cell or two, and they won't charge up properly, so you lose the lights after a fairly short time. A good battery should keep those lights on all night.
The 110VAC stuff (Microwave, A/C, and any outlets) will work with the shore power, and you may get some 110VAC from the outlets if your Inverter is working properly, and the batteries are fully charged.
Hope that helps some....but you may want to have the systems checked out by someone who knows RV electric a bit better.
Happy Camping:)
Bill & Claudia / DD Jenn / DS Chris / GS MJDogs: Sophie, Abby, Brandy, Kahlie, Annie, Maggie, Tugger & Beau ๐RIP: Cookie, Foxy & Gidget @ Rainbow Bridge.๐2000 Winnebago "Minnie" 31C, Ford V-10
Purchased April 2008:B FMCA# F407293
The Pets