Wow...that's almost a 'How far is up" type question... tough to answer...
You must have a converter/charger in your MoHo - most all RV's have em'...!
It's usually located near the
12 volt distribution panel with the fuses for the various circuits - I'd send out a search party and track it down...
When, plugged into shore power, check the output voltage - the converter side of the fuse panel would show the output voltage - with your batteries discharged, it should be above 14 volts, as the converter is pouring the coals to the batteries, trying to recharge em'....
Your furnace's blower motor uses more amps than any other item in a typical RV, so extended running during cold weather will discharge your batteries over night in many cases - and sooner with older batteries...
However, if you're plugged into shore power the furnace should be working, even if the batteries are discharged -
So it sounds like the converter charger isn't working - check AC circuit breaker for the converter - is it tripped...?
Is the battery disconnect switch functioning or in the 'disconnect'
position..?
Those are some quick ideas that come to mind - I'm a AS trailer guy, not a MoHo guy, but many of the basic 'systems' are similar...
If you have an onboard generator, it also could indeed be a problem with the transfer switch that changes power sources when hooked to shore power or generator...
Hope that helps...