Batteries or no batteries should NOT affect the 110 volt system. There's more going on here than a perhaps defective converter.
If the converter is humming you have 110 volts in there. If it's not getting to any other outlets start looking for tripped GFIs. Is the display on your microwave lit?
You say the batteries are good? Then there is a disconnect somewhere between them and the fuse box. The 12 volt side of the house should work whether the unit is plugged into 110 volts or not if the batteries are OK.
Keith J.
Sold the fiver and looking for a DP, but not in any hurry right now.