I am kinda confused a little here. If you are on shore power (either at the campground or house) there should be no parasitic battery drain, as the converters job besides being a battery charger...is to convert 120 volt AC to 12 volt DC, that then powers the 12 volt DC items (including the parasitic ones).
You should not run out of battery power overnight, if you are plugged into a form of 120 volt AC shore power, since the converter is providing power to the 12 volt items (ie furnace) INSTEAD of the battery.
When plugged into 120 volt shore power, your battery should not be depleting itself, unless it is the item providing power to the 12 volt items....hence your converter is not.
I would look to see if the converter is outputting (sounds like it is not) and make sure the circuit breaker for the converter is not tripped.
Mark Polk explains a converters function better than I do here:
RV Converters and Amp Draw - RV Information (RV Maintenance)