your post is confusing.
the microwave will only work with shore power or a generator. it's 110vAC and won't run on the batteries. neither will the a/c.
are you saying that even when plugged into shore power, your lights are dim and the furnace won't run?
if so, then either the converter is bad or the batteries are bad.
as was said, need more info, like year/make/model/type of RV you're in.
and Zoe the Wonder Dog(R.I.P.)
'07 Roo 23SS pushing an '07 Chevy Avalanche
Equalizer WDH and Prodigy BC